text:
Q: How to deploy laravel websockets on shared hosting? I have setup my laravel websockets on localhost it works fine on localhost but when i upload it to production server and gaves my the following error.
WebSocket connection to 'ws:<URL>/app/somekey?protocol=7&client=js&version=4.3.1&flash=false' failed: WebSocket is closed before the connection is established.

below is my config.php file
<?php

use BeyondCode\LaravelWebSockets\Dashboard\Http\Middleware\Authorize;

return [

    /*
     * Set a custom dashboard configuration
     */
    'dashboard' => [
        'port' => env('LARAVEL_WEBSOCKETS_PORT', 6001),
    ],

    /*
     * This package comes with multi tenancy out of the box. Here you can
     * configure the different apps that can use the webSockets server.
     *
     * Optionally you specify capacity so you can limit the maximum
     * concurrent connections for a specific app.
     *
     * Optionally you can disable client events so clients cannot send
     * messages to each other via the webSockets.
     */
    'apps' => [
        [
            'id' => env('PUSHER_APP_ID'),
            'name' => env('APP_NAME'),
            'key' => env('PUSHER_APP_KEY'),
            'secret' => env('PUSHER_APP_SECRET'),
            'path' => env('PUSHER_APP_PATH'),
            'capacity' => null,
            'enable_client_messages' => false,
            'enable_statistics' => true,
        ],
    ],

    /*
     * This class is responsible for finding the apps. The default provider
     * will use the apps defined in this config file.
     *
     * You can create a custom provider by implementing the
     * `AppProvider` interface.
     */
    'app_provider' => BeyondCode\LaravelWebSockets\Apps\ConfigAppProvider::class,

    /*
     * This array contains the hosts of which you want to allow incoming requests.
     * Leave this empty if you want to accept requests from all hosts.
     */
    'allowed_origins' => [
        //
    ],

    /*
     * The maximum request size in kilobytes that is allowed for an incoming WebSocket request.
     */
    'max_request_size_in_kb' => 250,

    /*
     * This path will be used to register the necessary routes for the package.
     */
    'path' => 'laravel-websockets',

    /*
     * Dashboard Routes Middleware
     *
     * These middleware will be assigned to every dashboard route, giving you
     * the chance to add your own middleware to this list or change any of
     * the existing middleware. Or, you can simply stick with this list.
     */
    'middleware' => [
        'web',
        Authorize::class,
    ],

    'statistics' => [
        /*
         * This model will be used to store the statistics of the WebSocketsServer.
         * The only requirement is that the model should extend
         * `WebSocketsStatisticsEntry` provided by this package.
         */
        'model' => \BeyondCode\LaravelWebSockets\Statistics\Models\WebSocketsStatisticsEntry::class,

        /*
         * Here you can specify the interval in seconds at which statistics should be logged.
         */
        'interval_in_seconds' => 60,

        /*
         * When the clean-command is executed, all recorded statistics older than
         * the number of days specified here will be deleted.
         */
        'delete_statistics_older_than_days' => 60,

        /*
         * Use an DNS resolver to make the requests to the statistics logger
         * default is to resolve everything to 127.0.0.1.
         */
        'perform_dns_lookup' => false,
    ],

    /*
     * Define the optional SSL context for your WebSocket connections.
     * You can see all available options at: http://php.net/manual/en/context.ssl.php
     */
    'ssl' => [
        /*
         * Path to local certificate file on filesystem. It must be a PEM encoded file which
         * contains your certificate and private key. It can optionally contain the
         * certificate chain of issuers. The private key also may be contained
         * in a separate file specified by local_pk.
         */
        'local_cert' => env('LARAVEL_WEBSOCKETS_SSL_LOCAL_CERT', null),

        /*
         * Path to local private key file on filesystem in case of separate files for
         * certificate (local_cert) and private key.
         */
        'local_pk' => env('LARAVEL_WEBSOCKETS_SSL_LOCAL_PK', null),

        /*
         * Passphrase for your local_cert file.
         */
        'passphrase' => env('LARAVEL_WEBSOCKETS_SSL_PASSPHRASE', null),
    ],

    /*
     * Channel Manager
     * This class handles how channel persistence is handled.
     * By default, persistence is stored in an array by the running webserver.
     * The only requirement is that the class should implement
     * `ChannelManager` interface provided by this package.
     */
    'channel_manager' => \BeyondCode\LaravelWebSockets\WebSockets\Channels\ChannelManagers\ArrayChannelManager::class,
];

this is my broadcasting.php file
<?php

return [

    /*
    |--------------------------------------------------------------------------
    | Default Broadcaster
    |--------------------------------------------------------------------------
    |
    | This option controls the default broadcaster that will be used by the
    | framework when an event needs to be broadcast. You may set this to
    | any of the connections defined in the "connections" array below.
    |
    | Supported: "pusher", "redis", "log", "null"
    |
    */

    'default' => env('BROADCAST_DRIVER', 'null'),

    /*
    |--------------------------------------------------------------------------
    | Broadcast Connections
    |--------------------------------------------------------------------------
    |
    | Here you may define all of the broadcast connections that will be used
    | to broadcast events to other systems or over websockets. Samples of
    | each available type of connection are provided inside this array.
    |
    */

    'connections' => [

        'pusher' => [
            'driver' => 'pusher',
            'key' => env('PUSHER_APP_KEY'),
            'secret' => env('PUSHER_APP_SECRET'),
            'app_id' => env('PUSHER_APP_ID'),
            'options' => [
                'cluster' => env('PUSHER_APP_CLUSTER'),
                'useTLS' => true,
                'host' => "127.0.0.1",
                'port' => 6001,
                'scheme' => 'http'
            ],
        ],

        'redis' => [
            'driver' => 'redis',
            'connection' => 'default',
        ],

        'log' => [
            'driver' => 'log',
        ],

        'null' => [
            'driver' => 'null',
        ],

    ],

];

and this is bootstrap.js file configuration
import Echo from "laravel-echo";

window.Pusher = require("pusher-js");

window.Echo = new Echo({
    broadcaster: "pusher",
    key: process.env.MIX_PUSHER_APP_KEY,
    wsHost: window.location.hostname,
    wsPort: 6001,
    disableStats: false,
    forceTLS: false
});

The code works completely fine on localhost but whenever i upload it to the shared hosting it didn't works. I am using laravel websockets for realtime notifications

A: All real-time technologies cannot be used with PHP shared hosting such as Bluehost Hostinger ...and that because of server limitations. All shared hosting servers are based on OnDemand-Service, if the client needs data has to request it from the service like a normal HTTP request, there is no continuous connexion between clients and the server

redpajama_set_name: RedPajamaStackExchange


Filming "That is How Motherhood Works"
What a year 2017 was! It was so full of joyful moments and it was so fun to capture many of them to share. Yet, I found that some of the most joyful moments couldn't be photographed. These moments were quiet, small, maybe even insignificant things that actually meant a whole lot to my life.


Located adjacent to I-675 on Highway 42, Metro Self Storage offers ground-level storage units for rent in Rex, Georgia. Our storage features include desirable amenities such as drive-thru access, climate-controlled units, and covered loading and unloading. Think of us as your Rex moving supplies headquarters and enjoy a one-stop-shopping experience for well-priced boxes and packing supplies for your next move or decluttering project.
If you are like most folks, tromping up and down stairs during a move is the last thing you want to do after a day of packing boxes. You can breathe a sigh of relief knowing your stored belongings are safely tucked away in ground level units when you choose our Rex storage facility.
With more than 40 years of storage experience you can count on Metro to bring you one of the best storage experiences in Clayton County. We offer residential and business storage solutions to several nearby neighborhoods and cities including Stockbridge, Morrow, Ellenwood, and Lake Spivey.
"The customer service is excellent! No problems, speedy attentive service with a warm welcoming spirit, I love being made to feel important when I..."
Life takes organization, and Metro Self Storage is here to help you with a simple storage solution. Our professionally trained and friendly storage team is qualified to meet your personal, recreational, and business storage needs. We welcome customers to our storage facility with sincerity and a warm smile and strive to provide the best storage experience possible.
With more than 40 years of self storage experience, Metro has perfected the art of self storage. We will help you find an excellent storage solution whether you need personal, business, recreational, or vehicle storage.


Like the third-year students on Wednesday, the children of I Cycle visited this Thursday, March 7 this important Museum that collects some works of our honored painter Inocencio Medina.
The activity, organized again by the company ARISTAS, was very entertaining and educational, performing leisure and artistic activities while the rest of the companions made the visit.
The teachers who accompanied the students in this case were their tutors: Marisol, Juana Mª, Puri and Begoña L. Martí, as well as Begoña L. Romero, Consuelo and Mª Pilar, together with the graduate student of the Teaching Degree, Alejandro.
Below we can see an extensive photo report made by the teachers.


text:
Grandparenting in divorced families
by Neil Ferguson, with, Gillian Douglas, Nigel Lowe, Mervyn Murch Neil Ferguson
Paperback(First Edition)
Hardcover-$111.95 Paperback-$47.95 View All Available Formats & Editions
Ship This Item — Qualifies for Free Shipping
Edition description:
6.14(w) x 9.21(h) x 0.02(d)
Neil Ferguson was a Research Associate attached to a Nuffield Foundation funded project in the Law School at Cardiff University. The project was co-directed by Professor Gillian Douglas, Professor Nigel Lowe, Professor Mervyn Murch and Dr Margaret Robinson.
Introduction; What do we already know about grandparents?; Grandparents' relationships with grandchildren:continuity and change; Activities with grandparents; Discipline and favouritism; The main grandparents; Grandparenting in divorced families: rights and policies; Communicating in divorced families; Taking sides; 'Being there': grandparents' financial, emotional and childcare support; Excluded grandparents; Conclusions: grandparents and family policy.
What People are Saying About This
"This highly topical book is one of the first studies of grandparenting in the UK. It provides a unique, detailed and often moving view of the roles that grandparents play in divorced families, and how their children and grandchildren respond to them. Its findings will be influential in shaping family policy, law, research and practice." Geoff Dench, Institute of Community Studies, London, UK.
"... a very interesting and readable book which, as intended, provokes a lot of thought about where grandparents fit in, not only in the lives of their grandchildren, but also in terms of social policy." Family Court Journal
"... a "must read" for those interested in the sociology of the family, as well as those deep in the trenches of family policy, research or practice." Family Matters "... it is important to acknowledge the book as a readable, stimulating and occasionally touching study. It is an important contribution to family studies." Ageing and Society

redpajama_set_name: RedPajamaCommonCrawl


OREANDA-NEWS. All companies in the food service business will have to keep credit records on product quality, which will be the basis of financing and other support for further development in Hebei.
The provincial government looked for ways recently to control the rapid growth of the food industry, where most of the suppliers are small and widely scattered. There were also cases of companies losing their credit and ethics in the food processing business and this has been a big concern for local people.
The new credit records will be consulted whenever a company applies for financial assistance, pays taxes, or looks for more or land or wants to expand the business. Any that fails to meet national standards or have a bad food safety performance record will be added to a blacklist.
To guarantee food safety, the government will do more to sort out companies that failed to meet safety and cleanliness rules for products and get tougher in punishing those responsible.
Holidays and large events with many people taking part will be a key focus of the food safety personnel to avoid accidents and the watchdog is asking local people to take part in the supervisory process and provide more information to reduce the use of unsafe food products.


text:
From Monument to Masses was a politically charged post-rock band which was founded in San Francisco in late 2000. They have released four albums to date, the latest full-length of which was released through Dim Mak on March 10, 2009. They played their final show on August 28, 2010 at the Great American Music Hall in San Francisco, California.

Overview
From Monument to Masses is an instrumental post-rock power trio composed of Francis Choung (drums, keys, programming), Sergio Robledo-Maderazo (bass, keys, samples), and Matthew Solberg (guitar, loops). From Monument To Masses is officially a "bi-coastal" band with two members living in the California Bay Area (Robledo-Maderazo, Solberg) and one residing in New York City (Choung). The band formed in 2000 in the San Francisco area, and eventually signed to Dim Mak Records in 2002. The band's first album, From Monument To Masses, was initially intended to be a demo recording to find a label; however, when Dim Mak's label owner Steve Aoki, who was also a long-time friend of Robledo-Maderazo from the Southern California Hardcore scene and his UCSB classmate, heard the recording, he signed the band and decided to release it as an eponymous debut LP.

The band name is a challenge to the "Great man theory" and a call for a paradigm shift. Rather than subscribe to the notion that "highly influential individuals" to whom monuments are built shape historical developments, the band name posits that it is actually the masses struggling for change who make history. In several interviews, that band has cited the American Civil Rights Movement as an example and note the hard work of hundreds of organizers that often gets overshadowed in history books by admittedly important individuals. The band members have often explained in interviews that their music is only one expression of their respective politics. Each member has been involved with various organizations throughout the life of the band, including the League of Filipino Students-SFSU, Challenging White Supremacy, Kalayaan School for Equity, BAYAN USA, and others.

Members
Francis Choung - drums, keys, programming
Sergio Robledo-Maderazo - bass, keys, samples
Matthew Solberg - guitar, loops

Discography

Albums
 From Monument To Masses (album)|From Monument To Masses (2002)
 The Impossible Leap in One Hundred Simple Steps (2003)
 Schools of Thought Contend (December 31, 2005)
 On Little Known Frequencies (March 10, 2009)

Singles
 "Beyond God & Elvis" (2008)

Influences
According to the band's MySpace website , the band culls their "sound" from several influences, some of which include:
Refused
Fugazi
Godspeed You! Black Emperor
DJ Shadow
At the Drive-In
Dead Prez
Tortoise
Radiohead
Led Zeppelin
Mogwai
Don Caballero
Rage Against the Machine
Digable Planets

References

http://www.alternet.org/wiretap/20945/
http://thesilentballet.com/dnn/Home/tabid/36/ctl/Details/mid/367/ItemID/2234/Default.aspx

External links
 Official website
 Dim Mak Records
 RCRD LBL page
 FMTM on YouTube
 Live videos at intimepop.com

Musical groups from San Francisco
American post-rock groups

redpajama_set_name: RedPajamaWikipedia


text:
Sell Your Car FREE
Driving the 759bhp Lamborghini Aventador SVJ
When I was 12-years-old my father crashed his Datsun in modest style. I was in the back seat and the front of the Japanese car folded origami-style as it connected with the iron fist of a Jaguar, driven by a man wearing yeti boots.
To be honest, it was almost certainly my Dad's fault – although he was in sensible shoes and the other bloke only had a pair of ridiculous fury feet to stand on. Names and numbers were exchanged – it later transpired the driver was the newspaper owner Eddy Shah.
I didn't know who Shah was, although years later I briefly worked for one of his newspapers where the security guard was also the arts critic. But I always remember the yeti boots and the fact he had a charisma to match his fancy footwear. My Dad wasn't impressed, neither was the insurance company.
You had to be a certain type of person to wear yetis back in the 1970s. David Bowie and Marc Bolan probably had several pairs, which was acceptable but a man in a random Jaguar in the suburbs of Manchester – what were the chances of that?
A stout pair of Church's or John Lobb's were standard issue for a chap in a Jag, exhaling through the door in a Fedora and a cloud of cigar smoke. But yeti boots? They haven't been seen in this parish for decades.
Which got me thinking about the type of footwear I would choose to drive the ultimate Lamborghini, the Aventador SVJ. If you haven't heard of it – or bought a Porsche GT2 RS thinking it would be the fastest production car on the planet – the SVJ broke the Nurburgring lap record with a time of six minutes, 44 seconds.
Piloted by Lamborghini factory driver Marco Mapelli, the track-focussed SVJ was more than two seconds faster than the GT2 RS. We know for certain that Signore Mapelli definitely wasn't in yeti boots, which is a shame because they are just the sort of flamboyant footwear an Aventador driver should wear and would certainly have annoyed the folk from Stuttgart. A lot.
Any Lamborghini from year dot – except for the rather pointless Urus SUV – is a car not to be driven in sensible shoes. Both the current Huracan and Aventador, for example, are total show offs, a cacophony of stage fireworks and sound effects not seen since the pyrotechnical, high heeled era of the Kiss concert.
So this hardcore SVJ version of the Aventador then promises to be more crazy, more astonishing and even more bonkers. I've polished up an old pair of flashy Ermenegildo Zegna trainers specially to drive it – and also because the footwells in any Aventador are a tad too snug for yeti boots.
The most sophisticated and powerful Lamborghini ever to leave the Sant' Agata production facility is more than likely a swansong for the company's titanic V12 unit too. A hybrid Lamborghini (aka Sian) has reportedly been shown to potential customers behind closed doors. To mark what could be a defining moment, the naturally aspirated block has been modified to the edge of reason for the SVJ.
The raft of changes designed to squeeze 759bhp from a 6.5-litre motor include titanium valves, redesigned cylinder head and a lighter flywheel. These help to reduce the weight of the enormous but low-slung SVJ by 50kg to a trim 1525kg – or not that much more than a bog standard Ford Focus.
Consequently, the SVJ revs to beyond 8,500rpm and offers a wider band of torque, served up through a permanent four-wheel drive system that has been tweaked for more rear-axle bias. I doubt many Lambo owners ever lift the rear-mounted, carbon-fibre engine cover to peep at the engine but it's exquisite. On paper, 0-62mph in 2.8 seconds and a top speed of 217mph match a car as menacing as a great white.
Perched on top is ALA 2.0 – the latest version of Lamborghini's active aerodynamic spoiler system that attracts attention like a radar beacon. The rear-wing increases downforce by more than 40 per cent compared to the previous Aventador SV, complemented by two self-adjusting flaps at the front of the car and one atop the engine lid.
It's ingenious stuff but requires a Venn diagram to explain properly. On a fast corner, the forces created by the rear wing can be deflected left or right too, helping to increase grip over the inside rear wheel where it is most needed.
While ALA deflects air brilliantly, it won't stop a trail of car spotters gathering in your wake, camera phones pressed to the windscreen. It's that sort of machine. At least the central pillar supporting the spoiler has been perfectly placed to block any visibility through the back screen. You might as well throw away the rear-view mirror and shave a few extra ounces off the kerb weight.
A blast from those sonorous twin tailpipes can send unwanted admirers scattering. A stab of the throttle unleashes a guttural snort like Brian Blessed suffering an asthma attack. There's no neighbour-friendly setting for those awkward, early morning starts either, so don't expect a barbecue invite from number 17.
But for sheer spectacle and larger than life appeal, the SVJ is the automotive equivalent of Concorde. It's also equally as tight inside, once the wing doors have been swung upwards to reveal a riotous mix of suede and leather. The bucket seats, borrowed from the Spanish Inquisition, are almost painful on a long journey, while visibility and headroom were an obvious afterthought.
There's nowhere to stash a phone let alone my spotted handkerchief and the eccentric dashboard layout was based on the switchgear in Dr Who's original Tardis. The flip-up cover over the starter button is borrowed from a fighter jet. It's a talking point for new passengers but has a knack of catching a shirt cuff at the most awkward moments.
At least Lamborghini has dispensed with those silly indicator buttons on the steering wheel itself. Impossible to operate at night-time, the tiny switches have been replaced with a conventional stalk off the column that's enormous, unsightly and clunky to operate.
There's nothing easy or straightforward about any Aventador – even climbing in and out is a Houdini-type feat designed to scalp the forehead of 6ft+ passengers. The sound system is so tinny it sounds like every DJ is playing vinyl, while luggage space under the front bonnet has to be supplemented with the passenger footwell.
Worst of all, the single clutch gearbox is so ridiculously antiquated it's almost comical. There's a fore and aft head-nodding workout at low speed, accompanied by cries of 'it's not my crap gearchange honest' but matters improve dramatically at speed, when the shift is smoother.
Straight line performance is simply epic, while heading into a corner, each high-revving downshift on those huge paddle-shifters is pure drama, especially on a wet British road in January. This amount of performance and grip takes some getting used to.
Supplemented by an old-fashioned V12 soundtrack and outlandish styling, the SVJ a sensory overload rarely found in any car these days. Even modern, efficient Ferraris and the clinical but soulless McLarens can't hold a candle to the vibrations and resonating thrills of the supersonic SVJ.
In any other circumstances it would be right to suggest this £350,000 Aventador is a two-seater that's reached its sell by date. But what Lamborghini has done here is turn an ageing supercar into something even more spectacular. Just for one last fling of the dice, for old times' sake – and it doesn't matter what shoes you want to wear either.
By enthusiasts, for enthusiasts.
Rolls-Royce Boat Tail – The Most Expensive Production Car Ever
First Drive: Aston Martin V12 Speedster – Roofless Roadster
Classics and Supercars © 2020.
Drive your advertising with
Classics and Supercars
Email us for further details


GoodTime is an automated interview scheduling platform that focuses on the convenience of candidates and the preference or suitability of the interviewer. Because aspects of the workflow are put into autopilot, recruiters can experience a simplified and accelerated interview management process. This begins with the appointment setup stage where you can select interviewers manually. Apart from that, you can also allow the solution to choose individuals with the optimal set of characteristics. Thereon, candidates can pick a time that is agreeable to them. Once they do, this synchronizes with the employers and talents' calendars alike to ensure that either party does not miss the set schedule.
GoodTime prides itself for providing a thoughtful set of features for interviewers and candidates alike. That is because the solution can revolutionize the entire interview scheduling and management process, as it considers the convenience of talents and recruiters alike. To that end, it allows candidates to select times and dates when they are available. They can do this from their desktops and mobile phones, thus enabling them to confirm their schedule wherever they are. Their choices then reflect on their and the employers' calendars, making certain that they do not miss their set appointment.
Aside from synchronizing calendar data, GoodTime also reminds interviewers and candidates about their schedule. It does this through email and SMS to doubly ensure that both parties remember the arranged meeting. Companies can also configure how often these alerts are sent, thus making sure that recipients receive only enough notifications.
The candidate's convenience is not the only the GoodTime considers when automating interview schedules. It also takes into account the well-being and preferences of recruiters. That is why the solution empowers interviewers to make known their scheduling preferences. The platform then considers this automatically when setting up interview schedules. Aside from that, GoodTime helps ensure that a company's recruiting staff does not experience fatigue to interview overload. As such, valuable employees remain productive and loyal to the company for a long time.
Aside from helping to arrange interviews, GoodTime also empowers recruiting and hiring teams to make their processes more efficient. It does this by providing smart insights about interview volume, cancellation rates, reschedule incidents, and many others. As such, HR units can strategize how they can level up accelerate their workflows to accommodate more talents. Thus, they can find suitable employees faster.
If you are interested in GoodTime it might also be a good idea to examine other subcategories of Best Applicant Tracking Software collected in our base of SaaS software reviews.
Since each organization has particular business needs, it is sensible for them to abstain from searching for a one-size-fits-all perfect software solution. Needless to say, it would be useless to try to find such a system even among widely used software platforms. The intelligent thing to do would be to jot down the various essential elements that require consideration such as main features, finances, skill levels of workers, company size etc. Then, you should do your groundwork thoroughly. Read some GoodTime reviews and investigate each of the other solutions in your shortlist in detail. Such in-depth homework can make sure you discard ill-fitting applications and select the solution that offers all the aspects you need for business success.
How Much Does GoodTime Cost?
GoodTime provides custom enterprise pricing. Contact the vendor to request a quote.
We realize that when you make a decision to buy HR Software it's important not only to see how experts evaluate it in their reviews, but also to find out if the real people and companies that buy it are actually satisfied with the product. That's why we've created our behavior-based Customer Satisfaction Algorithm™ that gathers customer reviews, comments and GoodTime reviews across a wide range of social media sites. The data is then presented in an easy to digest form showing how many people had positive and negative experience with GoodTime. With that information at hand you should be equipped to make an informed buying decision that you won't regret.
What are GoodTime pricing details?
What integrations are available for GoodTime?
If you need more information about GoodTime or you simply have a question let us know what you need and we'll forward your request directly to the vendor.


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Help me tap into the small business market. Tell me more about PMG's services, such as case design.


text:
\section{Introduction}
\setcounter{equation}{0}

{\bf Problem Formulation}
\vskip 0.2cm

A framework for many classical interpolation problems is as
follows. Given a set of distinct nodes $x_1,~\ldots~,~x_m$,
image points $y_1,~\ldots~,~y_m$ (not necessarily distinct)
and a family of functions $\mathcal{F}$, find whether there
exist functions $f\in\mathcal{F}$ so that
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:interp}
y_j=f(x_j)\quad\quad\quad\quad j=1,~\ldots~,~m.
\end{equation}
If yes, parameterize all of them, preferably within a degree
bound. There is a vast literature on the subject see e.g.
\cite{AA}-\cite{BoDy}, \cite{Ko}-\cite{YS}.
To simplify the discussion, we here focus on scalar real rational
functions. Thus degree simply means the maximum between the degree
of the numerator and of the denominator polynomials. The
polynomial (a.k.a. the Lagrange) interpolation \cite{La} (in
\cite{Mej} it is attributed to \cite{Wa}) is probably the best
known problem in this framework. For the case where $\mathcal{F}$
is the set of rational functions see \cite{ABKW} and if in addition
all functions in $\mathcal{F}$ are analytic in a disk of a
prescribed radius in $\mathbb C$, the problem was addressed in \cite{AA}.
\vskip 0.2cm

We shall denote by $\mathbb C_r$ $(\overline{\mathbb C}_r)$ the open (closed)
right half plane (the subscript stands for ``right"). The family of
functions $\mathcal{F}$ we here focus on, is of ~{\em positive
real},~ i.e. analytically mapping the open right half plane to its
closure. Namely, a real rational function $~f(s)~$ of a complex
variable $~s~$ is said to be positive if
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:DefPos}
{\rm Re}\left(f(s)\right)\geq 0
\quad\quad\quad\quad\forall s\in\mathbb C_r~.
\end{equation}
Interpolation problem with rational positive real functions
can be further classified by the domain the nodes
$x_1,~\ldots~,~x_m$ belong to.
\vskip 0.2cm

If the nodes $x_j$ are in $\mathbb C_r$, this amounts to the classical
Nevanlinna-Pick interpolation problem, see e.g.
\cite[Theorem 18.1]{BGR} and for real functions \cite{YS}. There,
from the interpolation data one constructs the
Pick matrix whose $j,k$ element is given by
\[
\frac{y_j^*+y_k}{x_j^*+x_k}
\quad\quad\quad\quad j,~k=1,~\ldots~,~m.
\]
It is known that there exist interpolating functions if and only
if the Pick matrix is positive semi-definite. Moreover, all
interpolating functions may be parameterized through this Pick
matrix. Recall that having the Pick matrix positive semi-definite
implies that each 2-dimensional minor is non-negative, which in turn
can be written as,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:NevPick}
\frac{|x_j-x_k|^2}{{\rm Re}(x_j){\rm Re}(x_k)}
\geq
\frac{|y_j-y_k|^2}{{\rm Re}(y_j){\rm Re}(y_k)}
\quad\quad\quad\quad m\geq j>k\geq 1.
\end{equation}
This condition means that the map from the nodes $x_1~,~\ldots~,~x_m$ to
the image points $y_1~,~\ldots~,~y_m$, is contractive in $\mathbb C_r$ in
the sense of Eq. \eqref{eq:NevPick}. This illustrates the fact that
the interpolation problem cannot be solvable for arbitray data set.
\vskip 0.2cm

\noindent
If the nodes are confined to the imaginary axis, an interpolation scheme,
elegant in its simplicity, appeared in \cite{ZehebLempel1966}.
\vskip 0.2cm

\noindent
If the nodes $x_j$ are in $\overline{\mathbb C}_r$ (with possibly some nodes
on $i{\mathbb R}$) the problem is much harder, see e.g. \cite[Chapter 21]{BGR},
\cite{BoCa}, \cite{BoDy}, \cite{Ko} and \cite{Sa}.
\vskip 0.2cm

\noindent
If the interpolation data is in whole plane, provided that
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:OddNevPick}
{\rm Re}(x_j){\rm Re}(y_j)
>0\quad\quad\quad j=1,~\ldots~,~m,
\end{equation}
one can still resort to the
classical Nevanlinna-Pick interpolation scheme: First, complete
the data set so that if $x, y$ is an interpolation pair,
then so is $-x, -y$. Then, from this extended data, take the $m$
nodes which are in $\mathbb C_r$, construct the the corresponding
Pick matrix and proceed as usual. Finally, use the fact (see
\cite{YS}) that whenever the Pick matrix is positive
semi-definite, among the interpolation functions there exists
some function with odd symmetry, i.e. $f(s)=-f(-s)~$ (a.k.a. Foster or
lossless functions, see e.g. \cite{Belev1968}, \cite{CL}, \cite{YS}).
If instead of the left and right half planes, $\mathbb C$ is partitioned
to the unit disk and its exterior, a similar idea is presented
in \cite[Section 5]{AA}.
\vskip 0.2cm

{\em In this work we focus on the case where the nodes
$x_1,~\ldots~,~x_m$ are all in $\mathbb C_l$ (the open left half
plane). We parameterize a large subset of rational positive real
interpolating functions whose degree is less or equal to
$m$. In particular, it is shown that this set is never empty.}
\vskip 0.2cm

\noindent
A key idea is the following: We construct two rational functions
sharing the same denominator: (i) an interpolating
function $p(s)$ (not necessarily positive real) and (ii) a strictly positive
real rational function ${\scriptstyle\Delta}(s)$ vanishing at the nodes.
Thus, for all $r\in{\mathbb R}$ the parametric rational function
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:KeyIdea}
f(s)=p(s)+r{\scriptstyle\Delta}(s),
\end{equation}
is interpolating. Moreover, for ${ r}$
``sufficiently large", $f(s)$ turns to be positive real.
\vskip 0.2cm

\noindent
Interestingly, we can mention two ideas conceptually similar to those
in the current work, which have appeared within completely different
interpolation frameworks: (i) The fact that for interpolation by low
degree rational functions, one should separately treat numerators and
denominators, appeared in the context of Schur functions in
\cite[Theorems 1, 2]{Georgiou1999}. (ii) In
\cite{AlpayLew2014} we have used Eq. \eqref{eq:KeyIdea}
for interpolation by structured matrix-valued polynomials.
For example, where $p(s)$ was an interpolating polynomial which on
$i{\mathbb R}$, attained Hermitian values, while the polynomial
${\scriptstyle\Delta}(s)$, vanishing at the nodes, was positive definite
on $i{\mathbb R}$, see \cite[Eq. (1.6)]{AlpayLew2014}.
\vskip 0.2cm

\noindent
In Section II we present a five steps interpolation procedure:
\vskip 0.2cm

\noindent
In Step 1 we parameterize all candidates for denominator polynomials
of the sought interpolating functions. Namely, all real polynomials,
of degree of at most $~m$, non-vanishing at the nodes.
\vskip 0.2cm

\noindent
In Step 2,
to each of these denominator polynomials we match a numerator to
obtain $p(s)$, a rational interpolating function (not
necessarily positive real).
\vskip 0.2cm

\noindent
In Step 3, we construct ${\scriptstyle\Delta}(s)$ strictly
positive real rational functions, vanishing at the nodes. We now
restrict the denominators of $p(s)$, the rational functions from
Step 2 to the subset of the resulting deniminators of
${\scriptstyle\Delta}(s)$.
\vskip 0.2cm

\noindent
To each of the resulting interpolating function $p(s)$, we
add $r{\scriptstyle\Delta}(s)$, a weighted version of the
strictly positive real rational functions, vanishing at the nodes
(which shares the same denominar). Thus,
$p(s)+ r{\scriptstyle\Delta}(s)$ is an interpolating function,
of degree of at most $m$. Furthermore,
for $r$ ``sufficiently large" it is positive real.
\vskip 0.2cm

\noindent
A closer scrutiny reveals that all interpolating, positive real
rational functions obtained, are so that the degree of the
denominator is larger or equal to the degree of the numerator.
\vskip 0.2cm

\noindent
In Step 5, we complete our the description of positive real
interpolating functions as follows: We repeat the previous
steps by constructing positive real interpolating functions from
the original nodes $~x_j$ {\em but to}\begin{footnote}{Consider
the case where $y_j=0$ for some $j$.}\end{footnote}~ $\frac{1}{y_j}$.
Finally, as the sought solution, we take the reciprocal of these
functions.
\vskip 0.2cm

\noindent
In Section III we illustrate the above procedure by detailed examples.
and add concluding remarks.

\section{A Recipe}
\pgfkeysvalueof{/tikz/dsp/label}}{sec:Recipe}

\subsection{Step 1: Constructing all real monic polynomials,
of degree ${\mathbf m}$ and ${\mathbf m-1}$, non-vanishing
at the nodes}
\pgfkeysvalueof{/tikz/dsp/label}}{SubSec:Denom}
\setcounter{equation}{0}
\vskip 0.2cm

\noindent
We shall do it in stages.
\vskip 0.2cm

\noindent
{\bf 1a}\quad Constructing all complex polynomials of degree of at
most~ $m$ ~with no roots at prescribed distinct points~
\mbox{$x_1,~\ldots~,~x_m\in\mathbb C$}.
\vskip 0.2cm

\noindent
First, denote by~ $\eta(s)$ ~the monic polynomial
(of degree~ $m$) ~whose roots are the prescribed distinct points~
\mbox{$x_1,~\ldots~,~x_m\in\mathbb C$},
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:eta}
\eta(s):=\prod\limits_{j=1}^m(s-x_j).
\end{equation}
Next, denote by~ \mbox{$\phi_1(s),~\ldots~,~\phi_m(s)$} ~the monic
divisors of $~\eta(s)~$ of degree $~m-1$, i.e.
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:phi}
\phi_j(s)=
{\scriptstyle
\frac{\eta(s)}{s-x_j}
}
=
\prod\limits_{\begin{smallmatrix}k=1\\k\not=j
\end{smallmatrix}}^m(s-x_k)\quad\quad\quad
j=1,~\ldots~,~m.
\end{equation}
\begin{La}\pgfkeysvalueof{/tikz/dsp/label}}{LemmaTilde}
For distinct $x_1,~\ldots~,~x_m\in\mathbb C~$ let
$\eta(s)$ and $\phi_1(x)~,~\ldots~,~\phi_m(x)$
be as in \eqref{eq:eta} and \eqref{eq:phi}, respectively.
\vskip 0.2cm

1. The set of all polynomials of degree of
at most $m-1$ can be parameterized by
\[
\sum\limits_{j=1}^mc_j\phi_j(s)
\quad\quad\quad\quad c_j\in\mathbb C.
\]
2. The set of all polynomials of degree of at
most $m$ can be parameterized by
\[
\tilde{d}(s)=b\eta(s)+\sum\limits_{j=1}^mc_j{\phi}_j(s)
\quad\quad\quad\quad b, c_j\in\mathbb C.
\]
3. If in addition,
\[
c_1{\cdots}c_m\not= 0,
\]
$\tilde{d}(s)$ forms the set of all polynomials of degree of at
most $m$ which do not vanish at \mbox{$x_1~,~\ldots~,~x_m$}.
\end{La}

{\bf Proof}\quad 1.
For each $j$, $j\in[1,~m]$, let
\[
{\phi}_j(s)=\sum\limits_{k=0}^{m-1}a_{jk}s^k
\]
be identified with
\[
a_j:=
\left(\begin{matrix}a_{jo}\\ a_{j1}\\ \vdots\\ a_{j,m-1}
\end{matrix}\right).
\]
Now by construction,
\[
\begin{split}
\begin{pmatrix}
1&x_1&x_1^2&\ldots&x_1^{m-1}\\
1     &x_2   &x_2^2 &\ldots&x_2^{m-1}\\
\vdots&\vdots&\vdots&\vdots&\vdots   \\
1     &x_m   &x_m^2 &\ldots&x_m^{m-1}
\end{pmatrix}
\begin{pmatrix}
a_{10}&a_{20}&\ldots&a_{m0}\\
a_{11}&a_{21}&\ldots&a_{m1}\\
a_{12}&a_{22}&\ldots&a_{m2}\\
\vdots&\vdots&\vdots&\vdots\\
a_{1,m-1}&a_{2,m-1}&\ldots&a_{m,m-1}
\end{pmatrix}&=\\
&\hspace{-3cm}
=\begin{pmatrix}
*&0&0&\ldots&0\\
0&*&0&\ldots&0\\
0&0&*&\ldots&0\\
\vdots&\vdots&\vdots&\vdots&\vdots\\
0&0&0&\ldots&*
\end{pmatrix}
\end{split}
\]
where $~*~$ denotes a non-zero element.
\vskip 0.2cm

The leftmost is a Vandermonde matrix, which is non-singular,
whenever \mbox{$x_1~,~\ldots~,~x_m$} are distinct, see e.g.
\cite[Item 0.9.11]{HJ}.
Thus, the middle matrix whose entries are $a_{jk}$ must
be non-singular, so in particular its columns form
a basis to $\mathbb C^m$. In other words
\mbox{${\phi}_1(s),~\ldots~,~{\phi}_m(s)$} form
a basis to all polynomials of degree of at most $m-1$.
\vskip 0.2cm

2. In a similar way, the coefficients of $\eta(s)$ in \eqref{eq:eta}
can be identified with a vector in $\mathbb C^{m+1}$. By adding a
bottom zero to each the above vectors $a_1~,~\ldots~,~a_m$,
they are embedded in $\mathbb C^{m+1}$. Thus, the problem is reduced
to verifying that the last element in the vector associated with
$\eta(s)$ is non-zero, but by  construction $\eta(s)$ is of
degree $m$, so indeed its bottom element is non-zero.
\vskip 0.2cm

3. Having no roots at \mbox{$x_1~,~\ldots~,~x_m$}. Note
that,
\[
{\tilde{d}}(s)_{|_{s=x_j}}=c_j{\phi}_j(s)_{|_{s=x_j}}=
\left(c_j\prod\limits_{\begin{smallmatrix}k=1\\k\not=j
\end{smallmatrix}}^m(x_j-x_k)\right)_{|_{c_j\not=0}}\not=0.
\]
In other words, having $~c_1{\cdots}c_m\not= 0~$ is
necessary and sufficient for these polynomials not to vanish at
the original points \mbox{$x_1~,~\ldots~,~x_m$}.
Thus the claim is established.
\qed
\vskip 0.2cm

In the sequel we focus on real polynomials and real rational functions.
Hence, we have the next stage.
\vskip 0.2cm

\noindent
{\bf 1b}\quad Assuming the prescribed distinct points
\mbox{$x_1,~\ldots~,~x_m\in\mathbb C$}, are closed under complex
conjugation, constructing all real polynomials of degree of at
most~ $m$, ~with no roots at these points.
\vskip 0.2cm

\noindent
Assume hereafter that, if necessary, the original set of
distinct points \mbox{$x_1,~\ldots~,~x_m\in\mathbb C$} ~is complemented
so it is closed under complex conjugation. Namely,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ComplexConjugation}
{\rm Im}(x_j)>0~\Longrightarrow~ x_{j+1}=x_j^*.
\end{equation}
Note that then in Eq. \eqref{eq:eta} the resulting~ $\eta(s)$ ~is
real.
\vskip 0.2cm

We now construct the sought polynomials.

\begin{La}\pgfkeysvalueof{/tikz/dsp/label}}{La:Denom}
For distinct~ $x_1,~\ldots~,~ x_m\in\mathbb C$, ~closed under complex
conjugation, let~ $\eta(s)$ ~and~
\mbox{$\phi_1(s),~\ldots~,~\phi_m(s)$} ~be as in Eqs.
\eqref{eq:eta} and \eqref{eq:phi}, respectively.
The set of all real polynomials $~\tilde{d}(s)~$ of degree, of at most
$~m$, with no roots at the
original points \mbox{$x_1,~\ldots~,~x_m$}, can be parametrized by
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:TildeD}
\tilde{d}(s)=b\eta(s)+\sum\limits_{j=1}^mc_j{\phi}_j(s)
\quad\quad
{\scriptstyle b}\in{\mathbb R}
\quad\quad
\begin{smallmatrix}
{\rm Im}(x_j)>0&~&0\not=&c_{j+1}=c_j^*
\\~\\
{\rm Im}(x_j)=0&~&0\not=&c_j\in{\mathbb R}.
\end{smallmatrix}
\end{equation}
\end{La}
\vskip 0.2cm

The claim follows from Lemma \ref{LemmaTilde} along with
Eq. \eqref{eq:TildeD}.
Note that in particular,~ $\tilde{d}(s)$ ~may have multiple roots.
\vskip 0.2cm

\noindent
In the sequel, the prescribed points~ $x_1,~\ldots~,~x_m$ ~will be referred
to as~ {\em nodes}.
\vskip 0.2cm

\noindent
Without loss of generality, we shall find it convenient to
distinguish in Eq. \eqref{eq:TildeD} between the cases $b=0$, $b\not=0$.
Furthermore to ease the distinction, we differently denote
the coefficients\begin{footnote}{Although as before,~~
$\begin{smallmatrix}
{\rm Im}(x_j)>0&~&0\not=&{\gamma}_{j+1}={\gamma}_j^*
\\~\\
{\rm Im}(x_j)=0&~&0\not=&{\gamma}_j\in{\mathbb R}.
\end{smallmatrix}$
}\end{footnote}
of the polynomials for $b=0$ and
$b\not=0$, i.e.
\[
\begin{matrix}
\tilde{d}_o(s)&:={\tilde{d}(s)}_{|_{b=0}}&=
\sum\limits_{j=1}^m{\gamma}_j{\phi}_j(s)
\\
\tilde{d}_1(s)&:={\tilde{d}(s)}_{|_{b\not=0}}&=
b\eta(s)+\sum\limits_{j=1}^mc_j{\phi}_j(s).
\end{matrix}
\]
We now can state the following.

\begin{Tm}\pgfkeysvalueof{/tikz/dsp/label}}{Tm:Denom}
For distinct nodes~ $x_1,~\ldots~,~ x_m\in\mathbb C$, ~closed under complex
conjugation, let~ $\eta(s)$ ~and~
\mbox{$\phi_1(s),~\ldots~,~\phi_m(s)$} ~be as in Eqs.
\eqref{eq:eta} and \eqref{eq:phi}, respectively. Let also~
${\gamma}_1,~\ldots~,~{\gamma}_m~$ and $~c_1,~\ldots~,~c_m~$
along with $~b\in{\mathbb R}$, be all non-zero parameters
as in Eq. \eqref{eq:TildeD}.
\vskip 0.2cm

The set of all real polynomials~ $d(s)$ ~of degree, of at most $~m$,
with no roots at these nodes, can be parametrized by two families,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:D}
\begin{matrix}
\tilde{d}_o(s)&=&\sum\limits_{j=1}^m{\gamma}_j{\phi}_j(s)&~&~&~
&~&
{\rm deg}\left(\tilde{d}_o(s)\right)&=&m-1
\\~\\
\tilde{d}_1(s)&=&b\eta(s)+\sum\limits_{j=1}^mc_j{\phi}_j(s)&~&~&~&~&
{\rm deg}\left(\tilde{d}_1(s)\right)&=&m.
\end{matrix}
\end{equation}
\end{Tm}
\vskip 0.2cm

\subsection{Step 2:
(Not necessarily positive real)
interpolating functions with prescribed denominator.}
\pgfkeysvalueof{/tikz/dsp/label}}{SubSec:Interp}

\noindent
With each of the denominators in Eq. \eqref{eq:D}, $d_o(s)$ and $d_1(s)$,
we here match a numerator, denoted by $\nu_o(s)$ and $\nu_1(s)$,
respectively, to obtain a rational (not necessarily positive real)
interpolating function (from $x_j$ to $y_j$).

\begin{Tm}\pgfkeysvalueof{/tikz/dsp/label}}{Tm:NotNecessarilyPosInterpolating}
Let the interpolation data in Eq. \eqref{eq:interp} be
closed under complex conjugation\begin{footnote}{Namely,
if ${\rm Im}(x)_j>0$ then $x_{j+1}=x_j^*$ and
$y_{j+1}=y_j^*$.}\end{footnote}, where the nodes
$x_1,~\ldots~,~x_m$ are distinct and in $\mathbb C_l$. Let also
the (non-zero) coefficients ${\gamma}_1,~\ldots~,~{\gamma}_m$
and $c_1,~\ldots~,~c_m$ be as in Eq. \eqref{eq:D}.
\vskip 0.2cm

\noindent
Construct the polynomials
(where $\phi_j(s)$ as in Eq. \eqref{eq:phi})
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:Nu}
{\nu}_o(s)=\sum\limits_{j=1}^my_j\gamma_j\phi_j(s)
\quad\quad
{\nu}_1(s)=\sum\limits_{j=1}^my_jc_j\phi_j(s)~.
\end{equation}
Then, the rational functions,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:DefP}
\begin{matrix}
\tilde{p}_o(s):=\frac{\nu_o(s)}{\tilde{d}_o(s)}
=\frac{\sum\limits_{j=1}^my_j{\gamma}_j\phi_j(s)}
{\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)}
&~&
\tilde{p}_1(s):=\frac{\nu_1(s)}{\tilde{d}_1(s)}
=\frac{\sum\limits_{j=1}^my_jc_j\phi_j(s)}
{b\eta(s)+\sum\limits_{j=1}^mc_j\phi_j(s)}~,
\end{matrix}
\end{equation}
(with $0\not=b\in{\mathbb R}$) interpolate between $x_j$ and $y_j$.
\end{Tm}
\vskip 0.2cm

\noindent
This result follows directly from the definition
of $\phi_j(s)$ in Eq. \eqref{eq:phi}.
\vskip 0.2cm

\noindent
We next construct additional
interpolating rational functions of degree of at most $~m$.

\begin{La}\pgfkeysvalueof{/tikz/dsp/label}}{La:TildeFunctionR}
Let $\eta(s)$, $d_k(s)$, $\nu_k(s)$, and $p_k(s)$, (with $k=0, 1$)
from Eqs. \eqref{eq:eta}, \eqref{eq:D}, \eqref{eq:Nu}, and
\eqref{eq:DefP} respectively.\\
The set of all real rational functions of degree $~m$, vanishing at the nodes
is given by,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:TildeDelta}
\begin{matrix}
\tilde{\scriptstyle\Delta}_o(s)&=\left(\frac{\tilde{d}_o(s)}{\eta(s)}\right)^{-1}&
=\left({\sum\limits_{j=1}^m\frac{{\gamma}_j}{s-x_j}}\right)^{-1}
\\~\\
\tilde{\scriptstyle\Delta}_1(s)&=
\left(\frac{\tilde{d}_1(s)}{\eta(s)}\right)^{-1}&=
\left(b+\sum\limits_{j=1}^m
{\scriptstyle
\frac{c_j}{s-x_j}
}
\right)^{-1}
\end{matrix}
\end{equation}
with $~{\gamma}_1,~\ldots~,~{\gamma}_m$, $~b$,
$c_1, ~\ldots~,~c_m$ all non-zero.
\vskip 0.2cm

\noindent
Using $\tilde{\scriptstyle\Delta}_k(s)$,
define the rational functions,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ParametricInterpolatingFuctionTildeF0}
\begin{matrix}
\tilde{f}_o(s)&:=&\tilde{p}_o(s)+{\scriptstyle r_o\tilde{\Delta}_o(s)}
=\frac{{\nu}_o(s)}{\tilde{d}_o(s)}+{\scriptstyle r_o}\frac{\eta(s)}{\tilde{d}_o(s)}
\\~\\~&=&
\frac{
{\sum\limits_{j=1}^my_j{\gamma}_j\phi_j(s)}+{\scriptstyle r_o}\eta(s)}
{\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)}
\end{matrix}\quad\quad{\scriptstyle
r_o\in{\mathbb R}~~{\rm parameter}},
\end{equation}
and
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ParametricInterpolatingFuctionTildeF1}
\begin{matrix}
\tilde{f}_1(s)&:=&\tilde{p}_1(s)+{\scriptstyle r_1\tilde{\Delta}_1(s)}
=\frac{{\nu}_1(s)}{\tilde{d}_1(s)}+{\scriptstyle r_1}\frac{\eta(s)}{\tilde{d}_1(s)}
\\~\\~&=&
\frac{\sum\limits_{j=1}^my_jc_j\phi_j(s)+{\scriptstyle r_1}\eta(s)}{b\eta(s)+
\sum\limits_{j=1}^mc_j\phi_j(s)}
\end{matrix}\quad\quad{\scriptstyle
r_1\in{\mathbb R}~~{\rm parameter}}.
\end{equation}
Then the following is true.

\begin{itemize}
\item[(i)~~~~]{}For arbitrary $~r_o, r_1\in{\mathbb R}~$ the functions $~\tilde{f}_o(s)~$
and $~\tilde{f}_1(s)~$ in Eqs.
\eqref{eq:ParametricInterpolatingFuctionTildeF0}
\eqref{eq:ParametricInterpolatingFuctionTildeF1}
are of degree of at most $~m$.
\vskip 0.2cm

\item[(ii)~~]{}For arbitrary $~r_o, r_1\in{\mathbb R}~$ the functions $~\tilde{f}_o(s)~$
and $~\tilde{f}_1(s)~$ in Eqs.
\eqref{eq:ParametricInterpolatingFuctionTildeF0}
\eqref{eq:ParametricInterpolatingFuctionTildeF1}
interpolate from $x_1,~\ldots~,~x_m$ to $y_1,~\ldots~,~y_m$.
\end{itemize}
\end{La}

{\bf Proof :}\quad
For convenience, throughout the proof, we omit the dependence on $k=0, 1$
and simply write $\tilde{\scriptstyle\Delta}(s)$,
$\nu(s)$, $d(s)$ and $f(s)$.
\vskip 0.2cm

\noindent
$(i)$ Recall that by construction (see Eqs. \eqref{eq:eta}, \eqref{eq:D}
and Theorem \ref{Tm:Denom}) the polynomials
$\eta(s)$ and $d(s)$ are relatively prime. Recall also
(see Theorem \ref{Tm:NotNecessarilyPosInterpolating}) that
\[
m={\rm degree}\left(\eta
\right)\geq
{\rm degree}\left(\nu
\right)=m-1.
\]
We thus have the following,
\[
\begin{matrix}
m&=&{\rm degree}\left(\tilde{\scriptstyle\Delta}\right)
\\~&=&
{\rm degree}\left({\scriptstyle\frac{\eta}{d}}\right)
\\~&=&
{\rm degree}\left({\scriptstyle d}\right)+
{\rm degree}\left({\scriptstyle\eta}\right)
\\~&=&
{\rm degree}\left({\scriptstyle d}\right)+
\max\left(
{\rm degree}\left({\scriptstyle\eta}\right),
{\rm degree}\left({\scriptstyle\nu}\right)
\right)
\\~&\geq&
{\rm degree}\left({\scriptstyle d}\right)+
{\rm degree}\left({\scriptstyle\nu+r\eta}\right)
\\~&\geq&
{\rm degree}\left(
{\scriptstyle\frac{\nu+r\eta}{d}}\right)
\\~&=&
{\rm degree}\left(
\tilde{\scriptstyle f}\right).
\end{matrix}
\]
$(ii)$ This is immediate from Theorem \ref{Tm:NotNecessarilyPosInterpolating}
along with the definitions of $\tilde{\scriptstyle\Delta}(s)$ and of
$\tilde{f}(s)$, see Eqs. \eqref{eq:TildeDelta},
\eqref{eq:ParametricInterpolatingFuctionTildeF0}
and \eqref{eq:ParametricInterpolatingFuctionTildeF1}, respectively.
\qed
\vskip 0.2cm

\noindent
Following Eq. \eqref{eq:interp}, we assume hereafter
that the data set is closed under complex conjugation and that
the distinct nodes are in the open left half plane, i.e.
\[
x_1,~\ldots~,~x_m\in\mathbb C_l~.
\]
Thus, some of the resulting interpolating functions in Eqs.
\eqref{eq:ParametricInterpolatingFuctionTildeF0}
and \eqref{eq:ParametricInterpolatingFuctionTildeF1} may be positive
real while other are not. Either way, the rest of this section is
depvoted to extracting positive real functions out of them. Specifically,
we shall devise a scheme of easily constructing large subsets of
positive real interpolating functions. This simplicity, based on the
structure of interpolating functions (see Corollary
\ref{Cy:ConvexInterpolation} below), comes on the expense of guranteeing
finding ~{\em all}~  positive real interpolating functions.

\subsection{Step 3:
All positive real rational functions of degree
$~{\mathbf m}$,
with prescribed denominator, and vanishing at the nodes.}
\pgfkeysvalueof{/tikz/dsp/label}}{SubSec:RationalFunctions}
\vskip 0.2cm

\noindent
We first recall well-known facts, which are fundamental to our construction.
\vskip 0.2cm

\begin{Tm}\pgfkeysvalueof{/tikz/dsp/label}}{Tm:Convexity}
The following is true.
\begin{itemize}
\item[(i)~~]{}For prescribed data, the family of interpolating rational
functions is convex.

\item[(ii)~]{}The set of rational positive real functions forms a convex cone.
\end{itemize}
\end{Tm}
\vskip 0.2cm

As a non-empty intersection of convex sets, is convex, one
can conclude the following.

\begin{Cy}\pgfkeysvalueof{/tikz/dsp/label}}{Cy:ConvexInterpolation}
For prescribed data set, whenever not empty, the family of rational
positive real interpolating functions, is convex.
\end{Cy}
\vskip 0.2cm

\noindent
We now resort to the following, see e.g.
\cite[Section 4.3]{CL},
\cite[Definition 6.4]{Khalil2000}.

\begin{Dn}\pgfkeysvalueof{/tikz/dsp/label}}{Dn:StrictPosDef}
{\rm
A rational function $f(s)$ will be called~
{\em Strictly Positive Real}
if $~f(s-\epsilon)~$ is positive real, for some~
$\epsilon>0$.
}
\end{Dn}
\vskip 0.2cm

\noindent
The following well known properties will be useful
in the sequel.

\begin{Tm}\pgfkeysvalueof{/tikz/dsp/label}}{Tm:SPR}
(i) If a rational function $~f(s)~$ is strictly positive real, then
\[
{\rm Re}\left(f(s)\right)>0
\quad\quad\quad\forall s\in\overline{\mathbb C}_r~.
\]
(ii) The set of positive real functions forms a Convex Invertible
Cone.
\end{Tm}
\vskip 0.2cm

\noindent
Item (i) follows from Definition \ref{Dn:StrictPosDef}
and for item (ii) see \cite[Proposition 4.1.1]{CL}.
\vskip 0.2cm

\noindent
Whenever the functions in Eq. \eqref{eq:TildeDelta} are strictly
positive real, the tilde will be omitted and they will be denoted by~
$~{\scriptstyle\Delta}_o(s)$ and $~{\scriptstyle\Delta}_1(s)$.
This is addressed next.

\begin{Ex}\pgfkeysvalueof{/tikz/dsp/label}}{Ex:PosRealCoefficients}
{\rm
We next illustrate the fact that for any set of nodes in $\mathbb C_l$, the open
left-half plane, one can choose the coefficients
$~{\gamma}_1,~\ldots~,~{\gamma}_m~$ and
$~c_1,~\ldots~,~c_m$,
so that in Eq. \eqref{eq:TildeDelta} one obtains~
{\em strictly positive real}
functions, ${\scriptstyle\Delta}_o(s)$ and ${\scriptstyle\Delta}_1(s)$.
\vskip 0.2cm

\noindent
As the reasoning is identical, we show it only for
$c_1,~\ldots~,~c_m$.
\vskip 0.2cm

\noindent
Specifically, if $x_j\in{\mathbb R}_-$ then $\frac{c_j}{s-x_j}$ is strictly
positive real for all $c_j\in{\mathbb R}_+~$.\\
If $~x_j\in\{\mathbb C_l\smallsetminus{\mathbb R}_-\}~$ (and from
Eq. \eqref{eq:ComplexConjugation} \mbox{$x_{j+1}=x_j^*$})
then taking \mbox{$c_{j+1}=c_j^*$} yields,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:DegTwoPosDefEx}
{\scriptstyle\frac{c_j}{s-x_j}}+{\scriptstyle\frac{c_j^*}{s-x_j^*}}
=
{\scriptstyle 2{\rm Re}(c_j)}\left(s+\frac{
{\scriptstyle
\left(
{\rm Re}(c_j)
\left(-{\rm Re}(x_j)\right)
+
{\rm Im}(c_j){\rm Im}(x_j)
\right)
}
s
+
{\scriptstyle
{\rm Re}(c_j)
|x_j|^2}
}
{
{\scriptstyle
{\rm Re}(c_j)
}
s+
{\scriptstyle
\left(
{\rm Re}(c_j)
\left(-{\rm Re}(x_j)\right)
-
{\rm Im}(c_j){\rm Im}(x_j)
\right)
}
}
\right)^{-1}.
\end{equation}
Thus, choosing $c_j$ so that,
\[
{\scriptstyle{\rm Re}(c_j)}>
{\scriptstyle\frac{|{\rm Im}(c_j){\rm Im}(x_j)|}{-{\rm Re}(x_j)}}\geq 0,
\]
is sufficient to guarantee that the function in Eq. \eqref{eq:DegTwoPosDefEx}
is strictly positive real.
}
\qed
\end{Ex}
\vskip 0.2cm

\noindent
Motivated by the above example we can state the following.
\vskip 0.2cm

\begin{Tm}\pgfkeysvalueof{/tikz/dsp/label}}{Tm:Delta}
Out of the polynomials $~\eta(s)$, $~\tilde{d}_o(s)~$ and $~\tilde{d}_1(s)~$
in Eqs. \eqref{eq:eta}, \eqref{eq:D} respectively, one can choose the
coefficients $~c_1~,~\ldots~,~c_m$, $b~$ and
$~{\gamma}_1~,~\ldots~,~{\gamma}_m~$ to construct all
monic\begin{footnote}{As in Eqs.
\eqref{eq:ParametricInterpolatingFuctionTildeF0}
\eqref{eq:ParametricInterpolatingFuctionTildeF1}
$~\tilde{\scriptstyle\Delta}(s)$ is scaled by $r$,
without loss of generality one can take $d(s)$
to be moinc.}\end{footnote} polynomials,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:Denominator}
\begin{matrix}
{d}_o(s)=&\sum\limits_{j=1}^m{\gamma}_j{\phi}_j(s)&
\sum\limits_{j=1}^m{\gamma}_j=1&
{\rm deg}\left({d}_o(s)\right)=&m-1
\\~\\
{d}_1(s)=&\eta(s)+\sum\limits_{j=1}^mc_j{\phi}_j(s)&~&
{\rm deg}\left({d}_1(s)\right)=&m,
\end{matrix}
\end{equation}
so that the rational functions from Eq. \eqref{eq:TildeDelta} are
strictly positive real, i.e.
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:Delta}
\begin{matrix}
{\scriptstyle\Delta}_o(s)=
\left(\frac{{d}_o(s)}{\eta(s)}\right)^{-1}=
\left(
{\scriptstyle\sum\limits_{j=1}^m\frac{{\gamma}_j}{s-x_j}}
\right)^{-1}
&\sum\limits_{j=1}^m{\gamma}_j=1
\\~\\
{\scriptstyle\Delta}_1(s)=\left(\frac{{d}_1(s)}{\eta(s)}\right)^{-1}=
\left(1+\sum\limits_{j=1}^m
{\scriptstyle
\frac{c_j}{s-x_j}
}
\right)^{-1}.
&c_1{\cdots}c_m\not= 0.
\end{matrix}
\end{equation}
Moreover:\\
The coefficients of $~{d}_o(s)$: ${\gamma}_1~,~\ldots~,~{\gamma}_m$ form
a convex set (excluding ${\gamma}_j=0$) within a
hyperplane in $\mathbb C^m$.\\
The coefficients of $~{d}_1(s)$: $c_1,~\ldots~,~c_m$ form a convex subset of
$\mathbb C^m$ (excluding $~m~$ hyperplanes $c_j=0$).
This set is positively unbounded in the sense that if
in Eqs. \eqref{eq:D}, \eqref{eq:Delta}
\[
c_1,~\ldots~,~c_m
\]
is an admissible set of parameters, then so is\begin{footnote}{
provided that, to preserve complex conjugation,
$c_{j+1}=c_j^*$ implies $\delta_{j+1}=\delta_j~$.}\end{footnote}
\[
c_1+\delta_1,~\ldots~,~c_m+\delta_m
\quad\quad\quad\quad
\delta_j\geq 0\quad j=1,~\ldots~,~m,
\]
Furthermore, the sets ${\gamma}_1~,~\ldots~,~{\gamma}_m$ and
$c_1,~\ldots~,~c_m$ can be
parameterized by a convex subset of ${\mathbb R}^m$ (excluding the axes).
\end{Tm}

{\bf Proof}\quad
From Example \ref{Ex:PosRealCoefficients} it follows that for arbitrary set of
nodes, this family of ${\scriptstyle\Delta}(s)$ functions is not empty.
\vskip 0.2cm

\noindent
To simplify establishing structural properties, we begin by
ignoring the condition that neither \mbox{${\gamma}_1,~\ldots~,~{\gamma}_m$}
nor $c_1,~\ldots~,~c_m$ vanish.
\vskip 0.2cm

\indent
By Corollary \ref{Cy:ConvexInterpolation}
the set of interpolating positive real rational functions is convex.
\vskip 0.2cm

\noindent
Indeed, if $c_1~,~\ldots~,~c_m$ and $\hat{c}_1~,~\ldots~,~\hat{c}_m$ are
two admissible sets in Eqs. \eqref{eq:D}, \eqref{eq:Delta} then
so is ${\scriptstyle\theta}{c_1}+(1-{\scriptstyle\theta})\hat{c}_1~
,~\ldots~,~{\scriptstyle\theta}{c_m}+(1-{\scriptstyle\theta})\hat{c}_m$,
for all ${\scriptstyle\theta}\in[0,~1]$.
\vskip 0.2cm

\noindent
The fact that $\sum\limits_{j=1}^m{\gamma}_j=1$ forms a hyper-plane
in $\mathbb C^m$ is straightforward. Next, to show that the set
$c_1,~\ldots~,~c_m$ is positively unbounded, one can resort again to
the construction in Example \ref{Ex:PosRealCoefficients}.
\vskip 0.2cm

\noindent
Recall however that the restriction that $\prod\limits_{j=1}^mc_j\not=0$,
implies that the set of coefficients $~c_1,~\ldots~,~c_m~$ forms
an almost convex cone, as it excludes $~m~$ hyper-planes, $~c_j=0$.
\vskip 0.2cm

\noindent
Real coefficients: Recall that  in \eqref{eq:D} the coefficients $c_j$ are
real or come in complex conjugate pairs. Specifically if
there are $q$ coefficients in the upper half plane and
$m-2q$ are real, they are described by a point in ${\mathbb R}^m$.
\qed
\vskip 0.2cm

\noindent
The above analysis suggests that in the coefficient space, it is enough
to find the {\em boundary} of the (almost convex) sets of admissible
${\gamma}_1,~\ldots~,~{\gamma}_m~$ and $~c_1,~\ldots~,~c_m$.
\vskip 0.2cm

\noindent
In the next step, we combine Theorems \ref{Tm:NotNecessarilyPosInterpolating}
and \ref{Tm:Delta}
to construct positive real interpolating functions of degree of at most $m$.
\vskip 0.2cm

\subsection{Step 4: Positive real
interpolating functions.}
\pgfkeysvalueof{/tikz/dsp/label}}{SubSec:PosInterp}

\noindent
To extract positive real functions, out of the set of interpolating
functions $\tilde{f}(s)$ in Eqs.
\eqref{eq:ParametricInterpolatingFuctionTildeF0} and
\eqref{eq:ParametricInterpolatingFuctionTildeF1},
we focus on those whose deniminator is given by Theorem \ref{Tm:Delta}.
This is formalized next.

\begin{La}\pgfkeysvalueof{/tikz/dsp/label}}{La:FunctionR}
Let $\eta(s)$, $d_k(s)$, $\nu_k(s)$, $p_k(s)$ and $~{\scriptstyle\Delta}_k(s)$,
(with $k=0, 1$) from Eqs. \eqref{eq:eta}, \eqref{eq:D}, \eqref{eq:Nu},
\eqref{eq:DefP} and \eqref{eq:TildeDelta} respectively.\\
Define the rational functions,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ParametricInterpolatingFuctionF0}
\begin{matrix}
{f}_o(s)&:=&p_o(s)
+
{\scriptstyle
r_o
{\Delta}_o(s)}
=
\frac{{\nu}_o(s)}{d_o(s)}
+
{\scriptstyle
r_o
}
\frac{\eta(s)}{d_o(s)}
\\~\\~&=&
\frac{
{\sum\limits_{j=1}^my_j{\gamma}_j\phi_j(s)}
+{\scriptstyle r_o}\eta(s)}
{\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)}
\end{matrix}\quad\quad{\scriptstyle
r_o\in{\mathbb R}~~{\rm parameter}},
\end{equation}
and
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ParametricInterpolatingFuctionF1}
\begin{matrix}
{f}_1(s)&:=&p_1(s)+{\scriptstyle r_1{\Delta}_1(s)}=
\frac{{\nu}_1(s)}{d_1(s)}+{\scriptstyle r_1}\frac{\eta(s)}{d_1(s)}
\\~\\~&=&
\frac{\sum\limits_{j=1}^my_jc_j\phi_j(s)+{\scriptstyle r_1}\eta(s)}
{b\eta(s)+\sum\limits_{j=1}^mc_j\phi_j(s)}
\end{matrix}\quad\quad{\scriptstyle
r_1\in{\mathbb R}~~{\rm parameter}}.
\end{equation}
Then, $~f_k(s)$ (and
$p_k(s)$) are interpolating function with $~{\scriptstyle\Delta}_k(s)$
strictly positive real, vanishing at the nodes (all sharing the same denominator).
\vskip 0.2cm

\noindent
Furthermore, the quantities,
\[
\begin{matrix}
-\inf\limits_{s\in{\mathbb C}_r}~\frac
{
{\rm Re}~p_o(s)
}
{
{\rm Re}~{\scriptstyle\Delta}_o(s)
}
&~&~&~&~&~&
-\inf\limits_{s\in{\mathbb C}_r}~\frac
{
{\rm Re}~p_1(s)
}
{
{\rm Re}~{\scriptstyle\Delta}_1(s)
}
\end{matrix}
\]
are well defined.
\end{La}

{\bf Proof :}\quad
For convenience, throughout the proof, we omit the dependence on $k=0, 1$
and simply write $\nu(s)$, $d(s)$ and $f(s)$.
\vskip 0.2cm

\noindent
The construction in Theorem \ref{Tm:Delta} guarantees that
in Eq. \eqref{eq:Delta}
\[
{\scriptstyle\Delta}(s)={\frac{\eta(s)}{d(s)}}
\]
is~ {\em strictly positive real} and thus by item (i) of
Theorem \ref{Tm:SPR}
\[
{{\rm Re}\frac{\eta(s)}{d(s)}}>0\quad\quad\quad
\forall{ s}\in{\mathbb C}_r~.
\]
Recalling that (i) the numerator $~\eta(s)$ vanishes only at $~m~$
points in ${\mathbb C}_l$, see Eq. \eqref{eq:eta} and (ii)
in addition ${\rm degree}(\eta)\geq{\rm degree}(d)$
see Eq. \eqref{eq:D}, in fact
\[
{{\rm Re}\frac{\eta(s)}{d(s)}}\geq\delta>0
\quad\quad\quad\forall{ s}\in{\mathbb C}_r~.
\]
Next, exploiting again the fact that $~{\scriptstyle\Delta}(s)~$ is
strictly positive real, see Theorem \ref{Tm:SPR}, implies that
$~d(s)~$ does not vanish in $\overline{\mathbb C}_r$. We can thus conclude
that
\[
\inf\limits_{s\in{\mathbb C}_r}~\frac
{{\rm Re}~p(s)}
{{\rm Re}~{\scriptstyle\Delta}(s)}
\]
is well defined, so the claim is established.
\qed
\vskip 0.2cm

\noindent
So far we have described interpolating rational functions $~f(s)~$
of degree of at most $~m$ whose denominator is so that
$~{\scriptstyle\Delta}(s)~$ is strictly positive real.
To proceed with the construction, the idea is very simple, see
Eq. \eqref{eq:KeyIdea}~:\\
With the same $~\eta(s)$, $~d(s)~$ and $~\nu(s)$ construct
the rational functions
\[
f_a(s):=
{\frac{\nu(s)}{d(s)}}
+
{
r_a
\frac{\eta(s)}{d(s)}}
\quad\&\quad
f_b(s):=
{\frac{\nu(s)}{d(s)}}
+
{ r_b\frac{\eta(s)}{d(s)}}
\]
where $r_a$ and $r_b$ are real parameters.
On the one hand, from Lemma \ref{La:FunctionR} it follows that $f_a(s)$
and $f_b(s)$ interpolate between with same data. On the other hand,
Theorems
\ref{Tm:SPR}
and
\ref{Tm:Delta}
imply that,
\[
r_a>r_b\quad\Longrightarrow\quad
{\rm Re}\left(f_a(s)\right)
>
{\rm Re}\left(f_b(s)\right)
\quad s\in\mathbb C_r~.
\]
Next, recall that by Eq. \eqref{eq:DefPos} $~f(s)~$ is positive real
whenever,
\[
{\rm Re}\left(f(s)\right)\geq 0
\quad\quad\quad\quad\forall s\in\mathbb C_r~.
\]
Thus, one can formally define
$\forall s\in\mathbb C_r$,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:DefUnderlineR}
\begin{matrix}
\underline{r}_o:=
{\rm arg}~\min\limits_{r_o\in{\mathbb R}}~{\rm Re}\left(f_o(s)\right)
={\rm arg}~\min\limits_{r_o\in{\mathbb R}}~{\rm Re}\left(p_o(s)
+r_o{\scriptstyle\Delta}_o(s)\right)\geq 0
\\~\\
\underline{r}_1:={\rm arg}~\min\limits_{r_1\in{\mathbb R}}~{\rm Re}\left(f_1(s)\right)
={\rm arg}~\min\limits_{r_1\in{\mathbb R}}~{\rm Re}\left(p_1(s)+
r_1{\scriptstyle\Delta}_1(s)\right)\geq 0.
\end{matrix}
\end{equation}
We next combine the above definition of $\underline{r}$ along with
Lemma \ref{La:FunctionR}.
\vskip 0.2cm

\begin{Pn}\pgfkeysvalueof{/tikz/dsp/label}}{Pn:UnderlineR}
Let the rational function $~f_k(s)~$ and the scalars
$~\underline{r}_k$ (with $k=0, 1$) be as in Eqs.
\eqref{eq:ParametricInterpolatingFuctionF0}, \eqref{eq:ParametricInterpolatingFuctionF1}
and \eqref{eq:DefUnderlineR}, respectively.\quad Then,
\[
\begin{matrix}
\underline{r}_o:=-\inf\limits_{s\in{\mathbb C}_r}~
\frac{{\rm Re}~p_o(s)}{{\rm Re}~{\scriptstyle\Delta}_o(s)}
&~&~&~&~&~&
\underline{r}_1:=-\inf\limits_{s\in{\mathbb C}_r}
\frac{{\rm Re}~p_1(s)}{{\rm Re}~{\scriptstyle\Delta}_1(s)}
\end{matrix}
\]
and $~f_k(s)~$ is positive real if and only if $~r_k\geq\underline{r}_k~$.
\end{Pn}

\noindent
{\bf Proof :}\quad For simplicity, we omit both the dependence on $~s~$
and the subscript $~k$. Using Eqs.
\eqref{eq:ParametricInterpolatingFuctionF0}
\eqref{eq:ParametricInterpolatingFuctionF1} note that
\[
{\rm Re}\left(f\right)={\rm Re}
\left(
{
\frac{\nu}{d}+r\frac{\eta}{d}}\right)
={\rm Re}\left(
{
\frac{\nu}{d}}\right)+r{\rm Re}\left(
{
\frac{\eta}{d}}\right).
\]
Now, $~f~$ is positive real if and only if
\[
{\rm Re}\left(f\right)\geq 0
\quad\quad\quad
\forall s\in{\mathbb C}_r~.
\]
Namely,
\[
r{\rm Re}\left({\frac{\eta}{d}}\right)\geq
-{\rm Re}\left({\frac{\nu}{d}}\right)
\quad\quad\quad
\forall s\in{\mathbb C}_r~,
\]
in turn, using the fact that ${\frac{\eta}{d}}$ is strictly
positive real, see Theorem \ref{Tm:Delta}, this means that
\[
r\geq
{\frac{-{\rm Re}~
\frac{\nu}{d}~
}
{~~~{\rm Re}~
\frac{\eta}{d}~
}}
\quad\quad\quad
\forall s\in{\mathbb C}_r~.
\]
Hence, one can conclude that $~f~$ in Eqs.
\eqref{eq:ParametricInterpolatingFuctionF0},
\eqref{eq:ParametricInterpolatingFuctionF1}
is
positive real, if and only if,
\[
r\geq
\sup\limits_{s\in{\mathbb C}_r}~
{\frac{-{\rm Re}~
\frac{\nu}{d}~
}
{~~~{\rm Re}~
\frac{\eta}{d}~
}}
=
-\inf\limits_{s\in{\mathbb C}_r}~
{\frac{{\rm Re}~
\frac{\nu}{d}~
}
{{\rm Re}~
\frac{\eta}{d}~
}}~,
\]
and by Eq. \eqref{eq:DefUnderlineR}, the proof is complete.
\qed
\vskip 0.2cm

\noindent
Noting that ${\rm deg}(\eta_k)=m-1$, for $k=0, 1$, while
${\rm deg}(\psi)=m$ together with the fact that
${\scriptstyle\Delta}_k$ is strictly positive real,
guarantees the following.

\begin{Ob}\pgfkeysvalueof{/tikz/dsp/label}}{Ob:rGeq0}
In Proposition \ref{Pn:UnderlineR},
\[
\underline{r}_o\geq 0\quad\quad\quad\underline{r}_1\geq 0.
\]
\end{Ob}

\noindent
Note that from Eqs.  \eqref{eq:ParametricInterpolatingFuctionF0}
and \eqref{eq:ParametricInterpolatingFuctionF1} it follows that
for $~r_k>0$, with $k=0, 1$ whenever there is no pole-zero cancelation,
the degree of the numerator of $~f_o(s)~$ or of $~f_1(s)~$ is $~m$.
Thus, all positive real interpolating functions $~f(s)~$ we have
constructed are of degree at most $~m$, but  {\em under the restriction
that the degree of the numerator is greater or equal to the degree of
the denominator}. In the next section we address the complementary
case where the degree of the
denominator is greater or equal to the degree of the numerator.

\subsection{Step 5: Additional positive real
interpolating functions.}
\pgfkeysvalueof{/tikz/dsp/label}}{SubSec:AllPosInterp}

Taking the original data, if one considers a function, say $g(s)$,
interpolating from \mbox{$x_1,~\ldots~,~x_m$} to\begin{footnote}{Assuming
$y_j\not=0$}\end{footnote}
\mbox{$\frac{1}{y_1}~,~\ldots~,~\frac{1}{y_m}$}, then $~\frac{1}{g(s)}~$
solves the original problem, where we have relied on the fact that the inverse
of a positive real function, is positive real, see item (ii) of
Theorem \ref{Tm:SPR}. Here are the details.
\vskip 0.2cm

\noindent
We follow the previous steps (while adding hat to the respective
functions) and first
mimic Theorem \ref{Tm:NotNecessarilyPosInterpolating}.

\begin{Tm}\pgfkeysvalueof{/tikz/dsp/label}}{Tm:HatNu}
Let the interpolation data\begin{footnote}{Assuming
$~y_j\not=0$.}\end{footnote} be as in Eq. \eqref{eq:interp}, the
(non-zero) denominator coefficients ${\gamma}_1,~\ldots~,~{\gamma}_m~$
and $~c_1,~\ldots~,~c_m~$ be as in Eq. \eqref{eq:D}.
Construct the polynomials
(where $\phi_j(s)$ as in Eq. \eqref{eq:phi})
\[
\hat{\nu}_o(s)=\sum\limits_{j=1}^m\frac{{\gamma}_j}{y_j}\phi_j(s)
\quad\quad\hat{\nu}_1(s)=\sum\limits_{j=1}^m
\frac{\hat{c}_j}{y_j}\phi_j(s).
\]
Then, the rational functions,
\[
\hat{p}_o(s):=\frac{\hat{\nu}_o(s)}{d_o(s)}
=
\frac
{\sum\limits_{j=1}^m\frac{{\gamma}_j}{y_j}\phi_j(s)}
{\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)}
\quad\quad
\quad
\hat{p}_1(s):=\frac{\hat{\nu}_1(s)}{d_1(s)}
=
\frac
{\sum\limits_{j=1}^m\frac{c_j}{y_j}\phi_j(s)}
{\eta(s)+\sum\limits_{j=1}^mc_j\phi_j(s)}
~,
\]
interpolate between $~x_j~$ and $~\frac{1}{y_j}~$ with
$~j=1,~\ldots~,~m$.
\end{Tm}
\vskip 0.2cm

\noindent
Note that indeed all the parameters are as before.
\vskip 0.2cm

\noindent
We next mimic Lemma \ref{La:FunctionR} and
construct the rational functions,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ParametricInverseInterpolationHatF0}
\begin{matrix}
\hat{f}_o(s)&:=&
\left(\hat{p}_o(s)+
{
\hat{r}_o{\scriptstyle\Delta}_o(s)}
\right)^{-1}
&~&{\hat{r}_o\in{\mathbb R}~~{\rm parameter}}
\\~\\~&=&
\left(
\frac{\hat{\nu}_o(s)}{d_o(s)}+
{
\hat{r}_o}\frac{\eta(s)}{d_o(s)}
\right)^{-1}
&=&
\left(
\frac{
\sum\limits_{j=1}^m\frac{{\gamma}_j}{y_j}\phi_j(s)
+\hat{r}_o\eta(s)}
{
\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)
}
\right)^{-1}
\end{matrix}
\end{equation}
and
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ParametricInverseInterpolationHatF1}
\begin{matrix}
\hat{f}_1(s)&:=&
\left(\hat{p}_1(s)
+
{
\hat{r}_1
{\scriptstyle\Delta}_1(s)
}
\right)^{-1}
&~&{\hat{r}_1\in{\mathbb R}~~{\rm parameter}}
\\~\\~&=&
\left(
\frac{\hat{\nu}_1(s)}{d_1(s)}+{\hat{r}_1}\frac{\eta(s)}{d_1(s)}
\right)^{-1}
&=&
\left(
\frac{
\sum\limits_{j=1}^m\frac{c_j}{y_j}\phi_j(s)
+{\hat{r}_1}\eta(s)
}
{\eta(s)+\sum\limits_{j=1}^mc_j\phi_j(s)}
\right)^{-1}
\end{matrix}
\end{equation}
where $\eta(s)$, $d_k(s)$, ${\scriptstyle\Delta}_k(s)$ and $p_k(s)$, with
$k=0, 1$ are ~{\em as before}, see Eqs.  \eqref{eq:eta}, \eqref{eq:D}
\eqref{eq:Delta} and \eqref{eq:DefP}, respectively.
\vskip 0.2cm

\noindent
As before, the rational function $~\hat{f}_k(s)~$ (with $k=0, 1$)
see Eqs.
\eqref{eq:ParametricInverseInterpolationHatF0},
and
\eqref{eq:ParametricInverseInterpolationHatF1}
interpolate
from $x_1,~\ldots~,~x_m$ to $y_1,~\ldots~,~y_m$
for all $~\hat{r}_k\in{\mathbb R}$. Out of this family, we next extract the
positive real subset. To this end,
we introduce the following notation,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:DefUnderlineHatR}
\begin{smallmatrix}
\underline{\hat{r}}_o:=
{\rm arg}~\min\limits_{\hat{r}_o\in{\mathbb R}}~{\rm Re}\left(\hat{f}_o(s)\right)
&=
{\rm arg}~\min\limits_{\hat{r}_o\in{\mathbb R}}~{\rm Re}\left(
\frac{\hat{\nu}_o(s)}{d_o(s)}+
\hat{r}_o\frac{\eta(s)}{d_o(s)}\right)^{-1}\geq 0
\\~\\
\underline{\hat{r}}_1:=
{\rm arg}~\min\limits_{\hat{r}_1\in{\mathbb R}}~{\rm Re}\left(\hat{f}_1(s)\right)
&=
{\rm arg}~\min\limits_{\hat{r}_1\in{\mathbb R}}~{\rm Re}\left(
\frac{\hat{\nu}_1(s)}{d_1(s)}+
\hat{r}_1\frac{\eta(s)}{d_1(s)}\right)^{-1}\geq 0
\end{smallmatrix}
\quad\quad\forall s\in\mathbb C_r~.
\end{equation}
\noindent
By using item (ii) of Theorem \ref{Tm:SPR},
we can next adapt Proposition \ref{Pn:UnderlineR}
to guarantee that the sought interpolating functions are indeed positive real.

\begin{Pn}\pgfkeysvalueof{/tikz/dsp/label}}{Pn:HatUnderlineR}
Let the rational function $~\hat{f}_k(s)~$ and the scalars
$~\underline{\hat{r}}_k(s)$ (with $k=0, 1$) be as in Eqs.
\eqref{eq:ParametricInverseInterpolationHatF0},
\eqref{eq:ParametricInverseInterpolationHatF1}
and
\eqref{eq:DefUnderlineHatR}, respectively.\quad Then,
\[
\begin{matrix}
\underline{\hat{r}}_o:=-\inf\limits_{s\in{\mathbb C}_r}~
\frac{{\rm Re}~\hat{p}_o(s)
}
{{\rm Re}~
{\scriptstyle\Delta}_o(s)
}
&~&~&~&~&~&
\underline{\hat{r}}_1:=-\inf\limits_{s\in{\mathbb C}_r}
\frac{{\rm Re}~\hat{p}_1(s)
}
{{\rm Re}~{\scriptstyle\Delta}_1(s)
}
\end{matrix}
\]
and $~\hat{f}_k(s)$ is positive real if and only if
$~\hat{r}_k\geq\underline{\hat{r}}_k~$.
\end{Pn}
\vskip 0.2cm

\noindent
We have shown that $~\hat{f}_o(s)~$ and $~\hat{f}_1(s)~$ are
positive real interpolating functions of degree at most $~m$,
where the degree of the numerator is larger or equal to the
degree of the denominator.
\vskip 0.2cm

\noindent
Simlar to the reasoning at end of Step 4, one can conclude
the following.

\begin{Ob}\pgfkeysvalueof{/tikz/dsp/label}}{Ob:HatrGeq0}
In Proposition \ref{Pn:HatUnderlineR},
\[
\underline{\hat{r}}_o\geq 0\quad\quad\quad\underline{\hat{r}}_1\geq 0.
\]
\end{Ob}

\section{Examples and Concluding remarks}
\setcounter{equation}{0}

The above recipe is illustrated through simple examples.
\vskip 0.2cm

\noindent
{\bf A.}\quad We start by illustrating the role of $~f_o(s)~$
vs. $~\hat{f}_o(s)$ in Eqs.
\eqref{eq:ParametricInterpolatingFuctionF0}
and \eqref{eq:ParametricInverseInterpolationHatF0}, respectively
to obtain interpolating functions having at $s=\infty$
either pole or zero.
\vskip 0.2cm

\noindent
{\bf (i)}\quad
Find a minimal degree positive real function $~f(s)~$ mapping
$x_1,~\ldots~,~x_m\in\mathbb C_l$ to
\mbox{$y_1=x_1,~\ldots~,~y_m=x_m$}.
Clearly the sought solution is
\[
f(s)=s.
\]
We now follow the above recipe and substitute in Eq.
\eqref{eq:ParametricInterpolatingFuctionF0}
\[
\begin{split}
f_o(s)&=\dfrac{{\sum\limits_{j=1}^my_j{\gamma}_j\phi_j(s)}
+r_o\eta(s)}
{\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)}\\
&=\frac{\sum\limits_{j=1}^mx_j{\gamma}_j\phi_j(s)+r_o\eta(s)}
{\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)}\quad\hspace{4.1cm}\mbox{\text{\rm for $y_j=x_j$}}\\
&=\frac
{\sum\limits_{j=1}^m
x_j{\gamma}_j\phi_j(s)+
\frac{r_o}{m}
\sum\limits_{j=1}^m
(s-x_j)
\phi_j(s)}
{\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)}
\\
&=
\frac
{
\sum\limits_{j=1}^m
\left(s+x_j(m\gamma_j-1)\right)
\phi_j(s)}
{m\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)}\quad\hspace{3.3cm}\mbox{\text{\rm for $r_o=1$}}
\\
&=s \quad\hspace{8cm}\mbox{\text{\rm for $\gamma_j\equiv\frac{1}{m}~.$}}
\end{split}
\]
\vskip 0.2cm

{\bf (ii)}
Find a minimal degree positive real function $~f(s)~$ mapping
\mbox{$x_1,~\ldots~,~x_m\in\mathbb C_l$} to
\mbox{$y_1=\frac{1}{x_1},~\ldots~,~y_m=\frac{1}{x_m}$}.
Clearly the sought solution is
\[
f(s)={\dfrac{1}{s}}~.
\]
We now follow the above recipe and substitute in Eq.
\eqref{eq:ParametricInverseInterpolationHatF0}
\[
\begin{split}
\hat{f}_o(s)&=\left(\frac{\sum\limits_{j=1}^m\frac{{\gamma}_j}{y_j}\phi_j(s)
+\hat{r}_o\eta(s)}{\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)}
\right)^{-1}
\\&
=
\left(
\frac
{\sum\limits_{j=1}^mx_j{\gamma}_j\phi_j(s)
+\hat{r}_o\eta(s)}
{\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)}
\right)^{-1}
\quad\hspace{3.1cm}\mbox{\text{\rm for $y_j=\frac{1}{x_j}$}}\\
&
=
\left(
\frac
{
\sum\limits_{j=1}^m
\left(s+x_j(m\gamma_j-1)\right)
\phi_j(s)}
{m\sum\limits_{j=1}^m{\gamma}_j\phi_j(s)}
\right)^{-1}
\quad\hspace{2.2cm}\mbox{\text{\rm for $\hat{r}_o=1$}}\\
&=\frac{1}{s}
\quad\hspace{8cm}\mbox{\text{\rm for $\gamma_j\equiv\frac{1}{m}~.$}}
\end{split}
\]
\vskip 0.2cm

\noindent
{\bf B.}\quad
Parametrize all positive real rational functions, of degree of
at most two, so that
\[
f(-1)=y_1\quad\quad\quad\quad f(-3)=y_2~,
\]
where $y_1, y_2\in{\mathbb R}$ are arbitrary.
\vskip 0.2cm

\noindent
First for reference, a direct computation reveals that all
rational functions, of degree of at most one, are given by
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ExReafDegreeOne}
f(s)=\frac{
{\scriptstyle\left(a(3y_2-y_1)+b(y_1-y_2)\right)}s+
{\scriptstyle 3a(y_2-y_1)+b(3y_1-y_2)}
}
{{\scriptstyle 2}({\scriptstyle a}s+{\scriptstyle b})}~.
\end{equation}
These functions are positive real whenever,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ExReafDegreeOneConstraints}
\begin{matrix}
a&\geq 0\\~\\
b&\geq 0\\~\\
a(3y_2-y_1)+b(y_1-y_2)&\geq 0\\~\\
3a(y_2-y_1)+b(3y_1-y_2)&\geq 0.
\end{matrix}
\end{equation}
The conditions in Eq. \eqref{eq:ExReafDegreeOneConstraints} may be
satisfied for all $~y_1,~y_2\in{\mathbb R}$ ~{\em umless},
$~0>y_1=y_2$.
\vskip 0.2cm

\noindent
This implies that for $y_1=y_2\geq 0$ there is a zero degree positive
real interpolating function, see item (iii) below. For $~0>y_1=y_2$,
the positive real interpolating functions are of degree of at least two,
see item (vi) below. In all other cases, there exist positive real
interpolating functions of degree one and above.
\vskip 0.2cm

\noindent
We now follow the recipe from the previous section.

From Step 1
\[
\eta(s)=(s+1)(s+3)=s^2+4s+3
\]
and
\[
\phi_1(s)=s+3\quad\quad\phi_2(s)=s+1.
\]
From Step 2, and using Eq. \eqref{eq:D} yields
\[
d_o(s)=s+\gamma\quad\quad\quad
{\gamma\in[0,~4]\smallsetminus\{1, 3\}},
\]
and
\[
{ d_1(s)}=
{
s^2+s(4+c_1+c_2)+3+3c_1+c_2}
\]
where $c_1$ and $c_2$ are such that
$ 1+{\frac{c_1}{s+1}}+
{\frac{c_2}{s+3}}$ is strictly positive real.
For $d_1(s)$ the set of admissible parameters is convex and
positively unbounded\begin{footnote}{From Theorem \ref{Tm:Delta}
it follows that in particular it contains the whole first quadrant
of the $\{c_2,c_1\}$ plane}\end{footnote}
(excluding the axes $c_1=0$ and $c_2=0$), it is given by
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:c1c2}
c_2>\left\{\begin{matrix}-3(c_1+1)&~&\frac{1}{8}\geq c_1\\~\\
-\frac{1}{3}(\sqrt{c_1}+{ 2\sqrt{2}})^2&~&c_1\geq\frac{1}{8}~.
\end{matrix}\right.
\quad\quad\quad c_1c_2\not= 0.
\end{equation}
\begin{figure}[ht!]
\begin{tikzpicture}[scale=1.2,cap=round]
\pgfplotsset{compat=1.10}
\pgfplotsset{every axis/.append style={
                   axis x line=middle,
                   axis y line=middle,
                   axis line style={->,color=black, thick},
                   xlabel={$c_1$},
                   ylabel={$c_2$},
              }}
     \begin{axis}[
             xmin=-5.1,xmax=35.3,
             ymin=-24.1,ymax=15.8,
             grid=both,
             ]
\addplot [domain=-5.0:0.125,samples=100,color=gray, very thin, name path=A]({x}, {12});
\addplot [domain=-5.0:0.125,samples=100,color=blue, thick, name path=B]({x},{-3*x-3});
\addplot [gray] fill between[of=A and B]

\addplot [domain=0.125:32,samples=1000,color=gray, very thin, name path=C]{12};
\addplot [domain=0.125:32,samples=1000,color=blue, thick, name path=D]({x},{-(x+4*(2*x)^0.5+8)/3});
\addplot [gray] fill between[of=C and D]
     \end{axis}
\end{tikzpicture}
\caption{\mbox{$c_1$, $c_2$ for $\frac{\eta}{d_1}$ strictly positive real,
Eq. \eqref{eq:c1c2}.
}}
\pgfkeysvalueof{/tikz/dsp/label}}{Figure:parameters}
\end{figure}

From Step 3
\[
\begin{matrix}
\frac{{\nu}_o(s)}{d_o(s)}&=&
{\frac
{
y_1(\gamma-1)
}{2}}
+
{\frac
{
y_2(3-\gamma)
}{2}}
+
{\frac
{(\gamma-3)(\gamma-1)(y_2-y_1)}
{2(s+\gamma)}}
&~&
{\gamma\in[0,~4]\smallsetminus\{1, 3\}}
\\~\\
\frac{{\nu}_1(s)}{d_1(s)}&=&\frac{(c_1y_1+c_2y_2)s
+3c_1y_1+c_2y_2}{s^2+s(4+c_1+c_2)+3+3c_1+c_2}&~&c_1,~c_2
\quad{\rm from~Eq.~\eqref{eq:c1c2}}.
\end{matrix}
\]
Now from Step 4
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ExNo}
f_o(s)=r_o(s+4-{\scriptstyle\gamma})+
{\scriptstyle\frac{y_1(\gamma-1)}{2}}+
{\scriptstyle\frac{y_2(3-\gamma)}{2}}
+(r_o+{\scriptstyle\frac{y_2-y_1}{2}})
\frac{\scriptstyle(\gamma-3)(\gamma-1)}{s+{\scriptstyle\gamma}}
\end{equation}
with $\gamma\in[0,~4]\smallsetminus\{1, 3\}$, and with ${ c_1,~c_2}$
from Eq. \eqref{eq:c1c2},
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ExN1}
f_1(s)=
\frac
{
{
r_1}s^2+
{
(4r_1+c_1y_1+c_2y_2)}s+
{
3r_1+3c_1y_1+c_2y_2
}
}
{
s^2+
{ (4+c_1+c_2)}s+
{ 3+3c_1+c_2}
}~.
\end{equation}
One can verify that taking $~r_o$, $~r_1~$ ``sufficiently large"
renders $f_o(s)$, $~f_1(s)~$ positive real.
\vskip 0.2cm

\noindent
Next, from Step 5, assuming $~{ y_1y_2}\not=0$
and $\gamma\in[0,~4]\smallsetminus\{1, 3\}$,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ExHatFo}
\hat{f}_o(s)=\frac
{\scriptstyle
{ 2y_1y_2}(s+\gamma)
}
{\scriptstyle
{ 2y_1y_2\hat{r}_o}s^2+
{
(8y_1y_2\hat{r}_o+(3-\gamma)y_1+(\gamma-1)y_2)}s
+{ 6y_1y_2\hat{r}_o+(3-\gamma)y_1+3(\gamma-1)y_2}}
\end{equation}
and with ${ c_1,~c_2}$
from Eq. \eqref{eq:c1c2},
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ExHatF1}
\hat{f}_1(s)=
\frac
{
s^2+
{
(4+c_1+c_2)
}
s+
{
3+3c_1+c_2
}
}
{
{
\left(\frac{c_1}{y_1}+\frac{c_2}{y_2}\right)}s+
{
\frac{3c_1}{y_1}+\frac{c_2}{y_2}
}
+
{
\hat{r}_1
(s+1)(s+3)
}
}
\end{equation}
Again, taking $~\hat{r}_o$, $~\hat{r}_1$ ``sufficiently large" renders
$~\hat{f}_o(s)$, $~\hat{f}_1(s)~$ positive real.
\vskip 0.2cm

\noindent
Here are five particular cases.
\vskip 0.2cm

\noindent
{\bf (i)}\quad
Recall that in the Introduction we pointed out that if $y_1, y_2\in{\mathbb R}_-$,
see Eq. \eqref{eq:OddNevPick}, one can still try to resort to the
classical Nevanlinna-Pick interpolation, seeking positive real odd
fuctions so that
\[
f(-1)=y_1\quad\quad f(1)=-y_1 \quad\quad f(-3)=y_2 \quad\quad f(3)=-y_2~.
\]
Now, the solvability condition in Eq.
\eqref{eq:NevPick} reads,
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:CondExOddNevPick}
{\frac{y_2}{y_1}}\in\left[
{\scriptstyle\frac{1}{3}}~,~3\right],
\end{equation}
and the resulting positive real odd interpolating functions
(of degree of at most two) are
\[
\begin{matrix}
g_a(s)&=&
\frac{8y_1y_2s}{(y_2-3y_1)s^2+3(y_1-3y_2)}
\\~\\
g_b(s)&=&
\frac{\left(y_1-3y_2\right)s^2+3\left(y_2-3y_1\right)}{8s}~.
\end{matrix}
\]
We now show, that these positive real odd functions, are special cases
of the above recipe:\\
Indeed, assuming the condition in Eq.
\eqref{eq:CondExOddNevPick} is staisfied, from Eq. \eqref{eq:ExN1}
\[
{f_1(s)}_{|_{c_1=\frac{4y_2}{3y_1-y_2}~~c_2=\frac{12y_1}{y_2-3y_1}~~r_1=0}}
=
g_a(s),
\]
and from Eq. \eqref{eq:ExHatF1}
\[
{\hat{f}_1(s)}_{|_{c_1=\frac{4y_1}{3y_2-y_1}~~c_2=\frac{12y_2}{y_1-3y_2}~~\hat{r}_1=0}}
=g_b(s).
\]
\vskip 0.2cm

\noindent
To further emphasize that our approach is different, in the four
following special cases (ii), (iii) and (v), the condition in Eq.
\eqref{eq:CondExOddNevPick} is not satisfied, so the classical
Nevanlinna-Pick interpolation is not applicable.
\vskip 0.2cm

\noindent
{\bf (ii)}\quad Take the special case where $y_1=1$ and $y_2=3$.\\
Clearly, $f(s)=-s$ is a real, anti-positive, minimal degree,
interpolating function. We next seek minimal
degree positive real interpolating functions.
\vskip 0.2cm

\noindent
Substituting these image points in $~f_o(s)~$ in Eq.
\eqref{eq:ExNo} yields the following positive real
interpolating functions,
\[
f_o(s)={ r_o}s+
{({r_o}+1)}\left(4-{\scriptstyle\gamma}
+\frac{\scriptstyle (\gamma-3)(\gamma-1)}
{s+{\scriptstyle\gamma}}\right)
\quad\quad
{\gamma\in[0,~4]\smallsetminus\{1, 3\}}.
\]
To guarantee minimal degree, further substitute
$~{\underline{r}_o}=0$, to obtain interpolating
functions with zero at infinity,
\[
f_o(s)=4-{\scriptstyle\gamma}
+\frac{\scriptstyle(\gamma-3)(\gamma-1)}
{s+{\scriptstyle\gamma}}
\quad\quad
{\gamma\in[0,~4]\smallsetminus\{1, 3\}}.
\]
Comparing with Eqs. \eqref{eq:ExReafDegreeOne} and
\eqref{eq:ExReafDegreeOneConstraints} reveals that in this case
our recipe yields all minimal degree (equals one) positive real
interpolating functions.
\vskip 0.2cm

\noindent
Similarly for $~\hat{f}_o(s)~$ in Eq. \eqref{eq:ExHatFo}
\[
\hat{f}_o(s)={\frac{s+{\scriptstyle\gamma}}
{
{\left(\frac{\gamma-1}{2}+\frac{3-\gamma}{6}\right)}s+
{
\frac{3(\gamma-1)}{2}+\frac{3-\gamma}{6}
}
+
{\hat{r}_o}(s+1)(s+3)}
}
\quad\quad\quad
\]
with $\gamma\in[0,~4]\smallsetminus\{1, 3\}$.
As before, to single out interpolating functions of degree one,
we focus on cases where $~\underline{\hat{r}}_o=0$. However,
then to guarantee positive realness, the range of the parameter
$~{\gamma}~$ needs to be further restricted, i.e.
\[
\hat{f}_o(s)=3\left({\scriptstyle\gamma}+
\frac{\scriptstyle(3-\gamma)(\gamma-1)}
{s+{\scriptstyle\gamma}}\right)^{-1}
\quad\quad
{\gamma\in\left[{\scriptstyle\frac{3}{4}},~4\right]\smallsetminus\{1, 3\}}.
\]
Here, at infinity, the interpolating function has neither pole nor zero.
\vskip 0.2cm

\noindent
Finally note that comparison with Eqs. \eqref{eq:ExReafDegreeOne}
and \eqref{eq:ExReafDegreeOneConstraints} reveals that in this case,
the recipe produced ~{\em all}~ interpolating functions of degree
one.
\vskip 0.2cm

\noindent
{\bf (iii)}\quad Take the special case where $y_1=y_2\geq 0$.\\
One can substitute in Eq. \eqref{eq:ExNo} $~\underline{r}_o=0~$
to obtain the minimal (=zero) degree interpolating
function \mbox{$f_o(s)\equiv y_1$}.\\
Similarly, one can substitute in
Eq. \eqref{eq:ExHatFo} $~\underline{\hat{r}}_o=0~$
to obtainn the minimal (=zero) degree interpolating
function \mbox{$\hat{f}_o(s)\equiv y_1$}.
\vskip 0.2cm

\noindent
{\bf (iv)}\quad
Take the special case where $0>y_1=y_2~$.\\
Recall that from Eqs. \eqref{eq:ExReafDegreeOne} and
\eqref{eq:ExReafDegreeOneConstraints} we know that there are no positive
real interpolating function of degree less than two.
\vskip 0.2cm

\noindent
To obtain interpolating functions use the recipe and
substitute in Eq. \eqref{eq:ExNo} to obtain,
\[
f_o(s)=y_1+r_o\frac{(s+3)(s+1)}{s+\gamma}
\quad\quad\underline{r}_o=\left\{
\begin{smallmatrix}
-y_1\frac{\gamma}{3}&~&~&\gamma\in(1,~3)\\~\\
-y_1\frac{1}{4-\gamma}&~&~&\gamma\in\{[0,~1)\cup(3,~4)\}.
\end{smallmatrix}
\right.
\]
Note that $\underline{r}_o$ turns to be unbounded, as $\gamma$
approaches 4.
\vskip 0.2cm

\noindent
{\bf (v)}\quad
Take the special case where $y_1=2$, $y_2=0$.\\
As before, substituting these image points
in Eqs.  \eqref{eq:ExReafDegreeOne} and \eqref{eq:ExReafDegreeOneConstraints}
(with $\frac{b}{a}={\scriptstyle\gamma}$)
reveals that {\em all}~ minimal degree (equals one) positive real interpolating
functions are of the form\begin{footnote}{Substituting in Eq. \eqref{eq:ExNo}
$r_o=0$, yields the subset of the interpolating functions
in Eq. \eqref{eq:ExZeroImageDegOne}, where
$4\geq{\scriptstyle\gamma}$}\end{footnote}
\begin{equation}\pgfkeysvalueof{/tikz/dsp/label}}{eq:ExZeroImageDegOne}
f(s)=({\scriptstyle\gamma}-1)\frac{s+3}{s+{\scriptstyle\gamma}}
\quad\quad\quad 3\not={\scriptstyle\gamma}>1.
\end{equation}
Next, address the case where the interpolating function is so that
the degree of the denominator is strictly larger then the the degree
of the numerator.
Now, recall that since the set of image points contains zero,
Step 5 of the recipe cannot be used. Nevertheless, all required
interpolating functions are obtained.
\vskip 0.2cm

\noindent
We start with a straightforward considerations: Since at $x=-1$, the
numerator is non-zero, but it vanishes at $x=-3$, it must be (at least)
of degree one. Thus, the denominator is (at least) of degree two.
Indeed, to obtain all minimal degree interpolating functions of the
required nature, substitute in Eq. \eqref{eq:ExN1}
\[
{f_1(s)}_{|_{r_1=0}}=
\frac{2c_1(s+3)}{(s+1)(s+3)+c_1(s+3)+c_2(s+1)}~,
\]
where adapting Eq.\eqref{eq:c1c2},
\[
0\not= c_2>\left\{\begin{matrix}-3(c_1+1)&~&c_1\in(0, \frac{1}{8}]\\~\\
-\frac{1}{3}(\sqrt{c_1}+{ 2\sqrt{2}})^2&~&c_1\geq\frac{1}{8}~.
\end{matrix}\right.
\]
\vskip 0.2cm

\noindent
{\bf C}.~
In the previous item the interpolation nodes were real.
We here illustrate the fact that the recipe is identical
for the non-real case, assuming the interpolation nodes
are closed under complex conjugation.
\vskip 0.2cm

\noindent
Assume that the interpolation nodes are
\mbox{$x_1=-\gamma+i\delta$} and
\mbox{$x_2=-\gamma-i\delta$} where $\gamma>0$ and $0\not=\delta\in{\mathbb R}$.
Hence,
\[
\eta(s)=(s+\gamma)^2+\delta^2
\]
We now construct the denominator polynomials.
\vskip 0.2cm

\noindent
Following Theorem \ref{Tm:Delta} a degree one numerator polynomial
$~d_o(s)~$ is given by the condition that the following rational
function is strictly positive real,
\[
\begin{matrix}
\dfrac{\eta(s)}{d_o(s)}
&=&\left(\dfrac{\scriptstyle\frac{1}{2}+i\beta}{s+\gamma+i\delta}+
\dfrac{\scriptstyle\frac{1}{2}+i\beta}{s+\gamma+i\delta}\right)^{-1}\\~\\~&=&
\dfrac{(s+\gamma)^2+{\delta}^2}{s+\gamma+2\beta\delta}\\~\\~&=&
s+\gamma-2\beta\delta+\dfrac{\scriptstyle\delta^2(1+4\beta^2)}{
s+\gamma+2\beta\delta}~,
\end{matrix}
\]
namely,
\[
\gamma>2|\beta\delta|.
\]
Hence one arrives at the following parametrization,
\[
d_o(s)=s+2\gamma(1-\theta)
\quad\quad\quad\quad\theta\in[0,~1).
\]
\qed
\vskip 0.2cm

{\bf Concluding remarks}
\vskip 0.2cm

\noindent
1.\quad As already pointed out in Corollary \ref{Cy:ConvexInterpolation},
for arbitrary prescribed data set in $\mathbb C$, the family of all
positive real interpolating functions is
convex (whenever not empty).
\vskip 0.2cm

\noindent
In contrast,
the set of rational functions of a degree of at most $m$ is a cone,
but highly non-convex. In fact, the degree of a sum of two rational
functions is higher than the degree of each of the summands,
unless one of the denominators divides the other.
\vskip 0.2cm

\noindent
When the interpolation nodes are in $\mathbb C_l$, the open left half plane,
we here introduce an easy-to-compute parametrization of positive real
interpolating functions as a subset of ${\mathbb R}^{2m+3}$, see item 2 for
details.
\vskip 0.2cm

\noindent
2.\quad For arbitrary interpolating data set in Eq. \eqref{eq:interp},
closed under complex conjugation, with nodes in $\mathbb C_l$, a large
subset of positive real interpolating functions of degree of at most
$m$ may be conveniently parametrized a union of convex subsets within
${\mathbb R}^{2m+3}$.
\vskip 0.2cm

\noindent
Indeed the coefficients in Eq. \eqref{eq:Denominator}
are so that $c_1,~\ldots~,~c_m$ form a positively unbounded convex subset
of ${\mathbb R}^m$, which in particular contains ${\mathbb R}_+^m$,
excluding the axes (see e.g. Figure \ref{Figure:parameters}). Next,
${\gamma}_1,~\ldots~,~{\gamma}_m$ form a hyper-plane in ${\mathbb R}^{m-1}$.
Finally, each of the four parameters $\underline{r}_o$, $\underline{r}_1$,
$\underline{\hat{r}}_o$,
$\underline{\hat{r}}_1$,
lies in $\overline{{\mathbb R}}_+$.
\vskip 0.2cm

\noindent
3.\quad Step 4 of the recipe relies on the fact that positive real
rational functions form a convex cone and that the set of interpolating
functions is convex. Steps 3 and 5 rely  on the fact that the set
of positive real rational functions is closed under inversion.
\vskip 0.2cm

\noindent
4.\quad The parametrization through
$f_o(s)$, $f_1(s)$, $\hat{f}_o(s)$, $\hat{f}_1(s)$
is motivated by ~{\em simplicity}. It is neither minimal, as
the same interpolation function may be obtained in
more than one way, see e.g. Example B(iii), nor is it comprehensive,
as some of the minimal degree interpolating functions
may be missing, see e.g. Example B(v).
\vskip 0.2cm

\noindent
5.\quad While the parametrization through $f_o(s)$, $f_1(s)$,
$\hat{f}_o(s)$, $\hat{f}_1(s)$ is convenient, focusing on~
{\em minimal degree} interpolating functions involves ``fine
tuning" of the parameters $~{\gamma}_1,~\ldots~,~{\gamma}_m$,
$~c_1, ~\ldots~,~c_m$, $~r_o$, $~r_1$, $~\hat{r}_o~$ and
$~\hat{r}_1$, see Examples A, B.
\vskip 0.2cm

\begin{center}
Acknowledgement
\end{center}

The authors thank Prof. V. Bolotnikov form the Math. Dept. at
the College of William and Mary, Williamsburg, Virginia, USA
for providing useful, constructive remarks at early stage of
this work.

redpajama_set_name: RedPajamaArXiv


Bananas are one of the oldest known cultivated plants, but were first grown in the United States in the 1880s, by entrepreneurs in early plantations in Jamaica. This new fruit was odd-looking, originally with seeds, and would grow only in very particular tropical climates. For years, the fruit was an unreliable product due to its short ripening period; storms at sea or delayed trains meant that these early banana salesmen would often open shipping crates full of rotten, unsellable fruit. But as advances in transportation and refrigeration shortened the time it took to bring bananas to market, they rose in popularity, cleverly marketed as a grocery staple, a fruit for the whole family.
However, the banana that people ate in the early 20th century was not the one we know today. There are hundreds of edible banana varieties, but to standardise production, banana companies selected a single type to grow: the Gros Michel, a large, flavourful banana. Gros Michel did well up until the 1950s. But then a fungus known as Fusariumwilt, or Panama disease, rapidly infected entire plantations, and caused a global collapse in the banana trade. The industry quickly found a replacement, a banana resistant to Panama disease, called the Cavendish. But while these new bananas were filling a growing Western appetite, Cavendish suffered from the same flaw that brought down Gros Michel: monoculture.


Recently Google has announced the retirement date of their Google Reader services. It seems since I have started blogging Google and their services have changed a lot. With this new change bloggers have been searching to find alternative services so readers can continue to follow posts and content. After all, we are nothing without readers and an audience.
I have decided to give Bloglovin' a try as a reader service. It is free to join and follow blogs or websites. You can use Facebook to register. Registration is literally two quick and easy steps.
Once you are signed in on Bloglovin' the feeds and blogs you are following are listed neatly on the left hand side and the majority of the screen displaying the posts in order from newest to oldest. You can scroll through to read content in an easy to read display, layout and format.
I have been logged on for about an hour now and I think this will be an excellent way for readers and subscribers to follow my blog.


Products available here are tailored toward campgrounds for selling in their camp stores. If you know a campground that you would like to see TORNADO™ fire starters in please submit their information.
Thank You! We hope to have our products available in their camp-store ASAP.


text:
How to increase your chances to get a promotion/reclassification in 2023?
19/01/2023 Events, Newsletter - 2023-01-27, Promotions, reclassification
We will put the January 2023 presentation here after 30.01.2023, in the meantime, the 2022 version is still available.
The promotion/reclassification process is not particularly user-friendly. It's not always clear how DGs choose who to reward, nor why some years appear to be better than others. There are situations suggesting that some staff are less favoured, while some staff are evaluated as 'not applicable', regardless of how well they work.
What happens between now and the publication of the draft promotion/reclassification lists in June? And what happens between then and the publication of the final lists in November?
As far as we interact with the system:
we all have a talk with our line manager (even those who have chosen to transferring the report from the previous year): here's our guide on how to prepare.
some appraisals will be less positive than expected, or even 'unsatisfactory': don't panic! Here's are some of the options.
we can appeal our non-inclusion in the draft June list
Our presentation will look at what happens behind the scenes: the work we don't see.
Attend our event
Monday, 30 January 2023 12.30-14.30
Here are the connection details
This presentation builds on our recent presentation on the self-assessment and covers:
the decision chain and the bodies involved,
your chances of getting a promotion/reclassification,
how to successfully file an appeal in June
The annual promotion and reclassification decisions are taken by the Appointing Authority (AIPN), on the basis of
proposals by each DG;
recommendations by a Joint Promotion (or reclassification) Committee (JPC).
DGs and JPCs must determine who deserves a promotion/reclassification on the basis of a comparison of merits. While all staff might take part in the appraisal exercise. Only those who are eligible for promotion/reclassification take part in the promotion/reclassification exercises.
Staff Appraisal Reward possible? Minimum time in grade ('seniority') to be eligible Average speed of reward 2023
instructions and timetable
Officials Yes Promotion 2 years Staff Regulations Annex I.B and our chart with those percentages converted to years Officials and temporary staff
Temporary agents (2a and 2d ) Yes Reclassification (JRC/GROW/RTD only?) [*] 2 years Staff Regulations Annex I.B Officials and temporary staff
Temporary agents (2b, 2c, 2e, 2f ) Yes no[**] Not applicable (N/A) N/A Officials and temporary staff
Contract Agents 3a Yes reclassification 2 years Page 3 of Ares(2021)2467829 – 12/04/2021 CSC note on CA 3a Contract Agents 3a
Contract Agents 3b Yes no ≤ 3 years N/A Contract Agents 3b
The comparison of merits on which the promotion procedure is based takes into account:
the appraisal reports since the last promotion or, failing that, since recruitment [***]
the use of languages in the execution of duties
the level of responsibilities exercised (to be taken into consideration on a systematic basis alongside the two other elements).
Unfortunately, the system is more than a little flawed and the system of quotas can have more of an influence on your advancement (or otherwise) than the (ostensible) comparison of merits.
Please remember also that you are evaluated on the time you were present: there should be no mention of your leave of any kind. It does not matter if you were off sick or on parental leave or on holiday, how good was your work while you were present?
For any of you who are concerned about the career development review (CDR) in the context of possible harassment, we appreciate the work of our colleagues in the Harassment Watch Network (HWN):
2021, Abuse of the annual appraisal exercise ("CDR") in the European Commission as tool of psychological harassment with recommendations how to prevent this and better protect the targetted colleagues: second analytical paper
2022, My Commission Calvary: Testimonies of harassment at the European Commission
If you appreciate our work, please consider becoming a member of Generation 2004.
[*] There is a joint committee on reclassification (not promotion) for Temporary Agents (TAs) of JRC/GROW/RTD only. We also see TAs with permanent contracts (yes, mental gymnastics are necessary here: to be permanently temporary) 2a and 2d listed for reclassification.
[**] 16.01.2023 HR response to our questions on TAs: Conditions of Employment of Other Servants (CEOS) (Article 15(2)) which states that only the provisions on annual reports [appraisal] from the Staff Regulations (SR) (Article 43) apply by analogy to TAs. By contrast, there is no such provision concerning promotion (SR Article 45). This means that, in principle, TAs are not subject to promotion. This principle is confirmed in the case-law of the General Court (e.g. T-366/15 P, paragraph 48).
[***] Probation reports cannot and should not be used for promotion purposes.
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