text:
Acielle/ Style Du Monde
Vogue's best looks from the Off-White fall/winter 2022 show
By Nicole Phelps
Titled Spaceship Earth, the Off-White fall/winter 2022 show was a celebration of the late Virgil Abloh's creative genius
Virgil Abloh was fashion's most frequent flyer, a multi-hyphenate creative director, DJ, architect, serial collaborator, and amplifier for the voices of the Black community. The man was not earthbound. So it makes a sort of sense that since his sudden passing last November we've been visited by a drone kite and runway angels.
After two tributes to Abloh at Louis Vuitton—the first in Miami just two days after he died and the second here in Paris last month—this posthumous Off-White show registered more as a celebration. A ride on "Spaceship Earth," as it was called. Among the passengers: Rihanna and A$AP Rocky, Idris and Sabrina Elba, Pharrell Williams and his family, the CEOs of Louis Vuitton and New Guards Group, and a who's who of designers including Jonathan Anderson, Olivier Rousteing, Matthew Williams, Guram Gvasalia, and Jerry Lorenzo.
The spaceship metaphor worked in more ways than one. This show included not just the Off-White fall/winter 2022 ready-to-wear collection, but also a new high fashion line (haute couture without the haute couture appellation) "designed by Virgil and completed by the creative teams and collaborators with whom he worked," the show programme explained. Abloh presented his fall 2021 collection during the Paris haute couture shows last July. At the time he said, "I get frustrated if I don't feel an evolution, and the message becomes monotonous." In retrospect, he may have been laying the groundwork for this new discipline.
But first up was the ready-to-wear. The show opened with snippets from an interview Pharrell Williams did in 2020. "Share the codes… share the cheat codes. A lot of us had to figure it out ourselves… that's where we go wrong," he said. "The more of us that learn the codes, the stronger we are." Arguably, code sharing was what went down on the runway, a sort of lesson in Abloh-isms from elevated hoodies to the mad pockets of his cargo gear to the mid-layer garments that made such waves on the red carpet to letterman jackets to the "quotes" branding he was so famous for. Here he saved his biggest statement for a pair of white flags printed with the phrase, "Question Everything."
And why not launch couture? Abloh wasn't at a loss for ideas, as the extensive Off-White fall/winter 2022 show programme reminded us. Each of the 28 high fashion looks was extensively annotated, only not with the number of hours for this embroidery or that handmade veil or train, but with little backstories. Which is far more meaningful. There was an ode to his friend Sarah Jessica Parker—The Carrie B.—complete with an enormous shopping bag. There was a self-portrait—The Verg—with an oversized jacket, baseball cap, sneakers, and sunglasses. There was a never worn 2020 Met Gala dress for Kylie Jenner. And there was a more serious message, it would seem, about the health struggle Abloh kept so intensely private, in Cindy Crawford's two handbags: one printed with MORE LIFE and another full of red and white pill capsules. The beats were provided by the DJ Jeff Mills, who, it can't be any coincidence, is known as The Wizard (both The Wizard of Oz and The Wiz were among Alboh's references at Louis Vuitton).
Making good on Pharrell Williams's point about sharing the codes, Off-White's parent company New Guards Group streamed tonight's show in 100 storefronts across Paris installed with TV monitors for the occasion: boutiques, barbershops, pharmacies, etc. At his last show before the pandemic in February 2020 Abloh said, "the ethos of Off-White is it's not just clothes. My inspiration and motivation is more the humanity level." Words to live by for whomever takes over the Off-White pilot seat.
Courtesy of Off-White
This article was originally published on Vogue.com.
Louis Vuitton Off-White Virgil Abloh Pharrell Williams fall/winter 2022
As I Am: Meet Vogue Singapore's March issue cover stars and models of past, present and future
By Maya Menon and Sandhya Mahadevan
Vogue's best looks from the Dolce&Gabbana fall/winter 2022 show
By Emily Heng
Vogue's best looks from the Prada fall/winter 2022 show
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The best bags seen on the fall/winter 2022 runways so far
By Nicole Ng


February 5, 2018, 1:14 p.m.
A black Lab mix showed up on a man's front porch in Lower Burrell, Pennsylvania, so he brought the friendly pup to Animal Protectors of Allegheny Valley, hoping the shelter could find the dog's owners.
The shelter workers found the dog's microchip but had to do a little investigating to see if they could find her owner. The first phone number they called had been disconnected, but they reached the dog's original vet, who said the dog was listed as deceased.
"We assured them that she was with us and was very much alive, and they provided us with a current number for her owners," shelter manager Gwen Snyder tells MNN.
They reached her owner, Debra Suierveld, who lives 10 miles from Apollo, where the dog was found.
"Her initial reaction was confusion," Synder says. "When [the medical coordinator] told her that her dog had been brought to us over the weekend, the owner told us her dog was right with her and had been all weekend."
But when they told Suierveld the dog's name, her confusion turned to shock. She was quiet for a moment, and told them that her dog, Abby, had been missing for 10 years. She disappeared one day when playing in the yard with her kids.
Suierveld rushed to the shelter to pick up her long-lost pup.
"Abby is a very sweet and friendly dog, so at first it was hard to tell if she recognized her owner, or if she was just being her sweet self," Synder says. "However, over the time the owner was here, Abby continued to move closer to her, sitting on her feet and leaning into her, so I do think her memory was coming back to her. She also remembered her name."
Abby is in very good health for her age, so shelter workers think someone must have been taking care of her. But it's likely they didn't think to ask a vet to scan her to see if she had a previous owner.
"We microchip all of our animals when they are adopted, and we believe microchips are an indispensable tool in reuniting lost pets with their owners," Snyder says. "Abby's reunion likely never would have happened if she'd not been microchipped."
A Pennsylvania family finds their lost dog who was gone for a decade.


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Gorillaz's 'The Now Now': Track by track breakdown of what is to come
With just days to go for the new Gorillaz album, here's a track-by-track breakdown of the whole thing, based on the numerous previews we've got so far.
By Suraj Prabhu
We're just a day away from a huge day for music releases in 2018. June 29 (Friday) will see new albums from the likes of Drake, Florence + The Machine, Ray Davies and of course, the much awaited 'The Now Now', the latest offering from Damon Albarn's virtual primate dub/rock/hip-hop crossover outfit, Gorillaz.
In our piece on what to expect from the album, we'd mentioned how it was uncharacteristic of Gorillaz to head straight into the studio for another album considering they dropped the massive, collaboration-overloaded 'Humanz' just last year, which received lukewarm reception from fans and mixed reviews from critics. Albarn revealed in a previous interview that he went ahead with 'The Now Now' not because he wanted to cash-in quickly, but mainly because it "gives him a reason to go on tour" and display some new material. Whatever be the case, we know so far that the 11-track effort, produced by James Ford (Arctic Monkeys, Depeche Mode) and Remi Kabaka, is a much trimmer album than 'Humanz', doing away with the long list of collaborators to instead focus more on Albarn's method to his madness.
We also have a very good idea of what's in store considering Gorillaz have practically given away the whole album beforehand, without caring much for an element of surprise, and I'm absolutely fine by that! Modern marketing gimmicks for album releases can be excruciatingly annoying these days and this is a welcome change. After debuting the Snoop Dogg and Jaimie Principle collaboration 'Hollywood' live in Chile, Gorillaz rolled out a total of five singles - 'Humility (ft. George Benson),' 'Lake Zurich,' 'Sorcererz,' 'Fire Flies' and then officially, 'Hollywood.' Meanwhile, 'Tranz,' 'Souk Eye' and 'Magic City' were premiered live at Germany's Rock AM Ring festival earlier this month. To top it all off, Albarn and company played the album in its entirety on June 24 at a gig in Japan and live-streamed the whole shebang via the Boiler Room.
So before the album drops and we have a chance to listen to the whole thing with the production sheen that Ford and Kabaka are sure to add, let's take a look at the tracklist of 'The Now Now' and break down the tracks to get a feel of what's to come. We'll do this by piecing together the vast array of previewed material from Gorillaz, including official single releases, concert footage and bootlegs.
'The Now Now' Tracklist:
1. 'Humility' (feat. George Benson)
2. 'Tranz'
3. 'Hollywood' (ft. Snoop Dogg and Jamie Principle)
4. 'Kansas'
5. 'Sorcererz'
6. 'Idaho'
7. 'Lake Zurich'
8. 'Magic City'
9. 'Fire Flies'
10. 'One Percent'
11. 'Souk Eye'
1. Humility (feat. George Benson)
The first of the previews was the album opener 'Humility', which features legendary jazz guitarist George Benson, one of only three collaborators on the entire album. As the sonic feel and the accompanying music video (featuring a very cheery Jack Black) showed, the album opener definitely sets us up for a breezy, beachside summer album. "Calling the world from isolation / 'Cause right now, that's the ball where we be chained," Albarn's alter ego 2-D drones, telling us that the lyrics of the album would also be more introspective, something that should anyway naturally follow with the lack of too many collaborators. The sound is a return to Plastic Beach with a breezy synth-pop style that contrasts nicely with Albarn's grim lyrical concerns. A low-tempo groove driven track, 'Humility' is a track that gets your feet moving and loosens up your shoulders for more to come.
2. Tranz
The dissonance between the merry synth-pop groove and the dour lyricism widens on 'Tranz'. As the title suggests, there is a tinge of psychedelia embedded into the sound as well as the theme of the song. Albarn's vocals (at least live in Japan) are instantly reminiscent of Ian Curtis and the sound isn't too far from early Joy Division as well. The song is about 2-D returning from a heavy Saturday night of partying to find himself in a 'Tranz' state. "When you get back on a Saturday night / And your head is caving in / Do you look like me, do you feel like me / Do you turn into your effigy?" the chorus goes before exploding into a full-fledged psychedelic disco experience.
3. Hollywood (ft. Snoop Dogg and Jamie Principle)
The first of several passports stamps from the album (Track 4 is 'Kansas', track 6 is 'Idaho') Hollywood is the only other song apart from the album opener 'Humility' that sees a collaboration. This time from rap legend Snoop Dogg and the pioneer of the house genre in USA, Jaimie Principle. At 4 minutes & 53 seconds, it's the longest track on the album, mainly to allow elbow room for collaborators Snoop and Principle to flex their stuff. The song is a love-hate letter to the personification of Hollywood, all her excesses as well as her atrocities. Snoop's guest verse echoes this, but it's definitely far from his best cameos.
The second passport stamp comes in the form of the soon-to-be fan favorite 'Kansas'. Arguably the most accessible and pop-leaning song from the album, 'Kansas' has a simply constructed melody and an even simpler, super catchy chorus. The recurring line "I'm not gonna cry" almost begs to be sung along. The most distinct element of the song, of course, is the bassline, which sounds like it's coming from one of Peter Frampton's talk boxes, only with the bass turned way up!
5. Sorcererz
Bang in the middle of the album comes 'Sorcererz'. The synth assault gets heavier by now, underlined by damp funk rhythms and 2-D's croaky vocals. The song comes across a little like an instruction manual to let go of your worries, but the constant groove of the drum and bassline make it less instruction manual like and more of a yoga teacher at a hippie-filled music festival. "Everybody cool down / Everybody see yourself", "Everybody mining their own inner visions," 2-D sings, giving listeners some time to cool off before the second half of the album.
6. Idaho
The third and final ode to a geographical location on the album comes in the form of 'Idaho', which does a complete left turn from the first half of the album. The synths are replaced with jangly, folksy guitars as Albarn and 2-D belt out a pining, crooning love-letter to the state of Idaho. The chorus might tell us a little about the songwriting behind the song: "Idaho, Idaho /There's a beauty on the road / And everyday I look out of the bus / Silver linings getting lost." It appears that Albarn wrote the song on the 'Humanz' tour, probably on the tour bus, something that he's done before. Gorillaz fans will remember the lo-fi surprise album, 2010's 'The Fall', which was entirely recorded on Albarn's iPad while on the road, and comparisons between 'The Now Now' and 'The Fall' are sure to follow, although it's safe to say that 'The Now Now' is definitely a more fully-formed, focused album.
The brief interlude on Idaho immediately makes way for more funk and more heavy synths on 'Lake Zurich', which was the second single to be previewed from the album. The only instrumental effort on the album, it's a track that's pretty close sonically to French electronic duo Justice. The track could have served as an interlude for the final part of the album but at 4 minutes plus, it's more fully formed with subtle treats thrown into the electronic mix.
8. Magic City
Probably the most morose song of the album 'Magic City' is also the most brooding and introspective. The lyrics blur the lines between Albarn and his alter ego 2-D, as we see a glimpse of a self-aware singer alienated from humanity, aching to come back and join the fun during a time when we've evolved far enough to put a "billboard on the moon".
9. Fire Flies
The talk box like bassline returns on 'Fire Flies', courtesy of Gorillaz's new bassist Ace (who many will remember from the Gang Greene Gang on The Powerpuff Girls). As the story goes, regular bassist, Murdoc Nicols is in jail after getting involved in some sketchy stuff with a kingpin of a crime family! Another low-tempo breezy summer track like the album opener, 'Fire Flies' is a desperate love song of loss, fear and resentment. The line that embodies the song best is the grim warning: "And if you say goodbye too many times / The sentinels will find me and switch me off this time"
10. One Percent
The final part of the album is surely more somber than the poppier first half. The severely stripped-down 'One Percent' falls in line with the mellower tones of the album, the sparse vocals giving way to a melancholic build of eclectic atmospheric sounds. The brooding introspection continues, but while it was earlier hidden underneath a blast of synth-pop, it now lies bare for all to see.
Wrapping up the album with the third consecutive love song, 'Souk Eye' is a curious number that starts off with a skittering beat and soft acoustic guitars over shuffling maracas, but steadily builds into a grandiose romantic number, peppered with twangy synths and a rumbling drum arrangement. Sonically it's pretty far from the breezy album opener 'Humility', and overall, it holds its own as a close to a pretty well-rounded album.
The Verdict: "The Now Now', as the name suggests has a sense of urgency to it, but it is definitely much more well-rounded than Gorillaz's other urgent release 'The Fall'. The sound is closer to 'Plastic Beach' and the album is clearly divided into two halves with the first being more up-tempo and dancier than the latter half. If you're expecting the heydays of 'Demon Days', you'll obviously leave disappointed, but singles like 'Tranz', 'Kansas' and 'Sorcererz' have the potential for some heavy rotation on radio play. All said and done, it's a refreshing change from the bloated 'Humanz' and offers a much trimmer, tighter and more introspective side to Albarn.

redpajama_set_name: RedPajamaCommonCrawl


Here's the way I used to handle default parameters.
Much cleaner, especially as you add more parameters with their own defaults. However this just left me wondering about how far you could take these default parameters. Are they bound when the function is defined, or when it is called? What kind of expressions can you set as the value?
That looks promising. This answered my first question, parameters are bound when the function is called, not when it is defined. That's super handy. This also handles falsy values porperly, good stuff.
Babel has an additional mode called High Compliancy. This adds some extra noise to the above example to handle temporal dead zones. Luckily we don't have to worry about that with these examples.
I also noticed that FireFox already has default parameters implemented. So I opened up FireFox's developer console and did similar tests. Everything worked just as you would expect based on the Babel code.
Just to be sure there wasn't a bug in here I went ahead and dove into the ES6 Spec for Function Definitions. It looks like FireFox and Babel got it right. Default parameters are late binding and allow for basically any expression.
I'm glad we have Babel so we can use these great features now.


Reporting the findings of data-driven discovery is often thought of as "the crucial last step" in an analytics program. Unfortunately, this is often misinterpreted as meaning it is of lesser importance – nothing could be further from the truth!
Dr Stephanie Evergreen is the author of Presenting Data Effectively: Communicating Your Findings for Maximum Impact. She is a well-known expert on matters involving visualization and communications, and she spoke to me to pass on some basic, but often overlooked, tips on presenting data so it will make an impact.
"One of our strongest storytelling tools is to use grey plus some kind of action color", Evergreen tells me. Anyone used to seeing corporate reports will be used to multicolour pie charts and graphs, often using whatever colors come up by default in a template.
This is missing an opportunity to take advantage of the storytelling potential of color – the primary component our sense of sight is based on.
"Our eyes are attracted to color – we're built that way – so what we should be doing when we're telling stories with data is some kind of action color that will drive the eyeballs towards the part of the data where the story is being told.
Great visualizations don't leave a lot of room for words – the point is to tell the story using images. That doesn't mean words aren't important – the smart strategy is to think of it as meaning that every letter is essential, and there's certainly no room for wasted words.
Numbers do a great job of filling our rational need for quantifiable information. But when it comes to communicating how things will actually impact our real lives, or affect the world we live in, some form of humanizing or grounding the data in reality is often effective.
"Sometimes that can be achieved by putting a photograph which relates to your topic next to your graph. Sometimes it can just be as simple as including a graph." Anything besides a table of raw numbers makes assessing the true, real-world implications of data an easier task.


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\section{Introduction}\label{sec:intro}

There are two fast developments in the past years regarding products and sums of random matrices. One concerns the macroscopic level densities that can be elegantly computed via the free probability  approach~\cite{Speicher2011} introduced by Voiculescu~\cite{Voiculescu1991}. The other development has the local spectral statistics in its focus for which the concept of polynomial ensembles~\cite{KZ2014} has proven fruitful, see also Definition~\ref{def:ensembles}. The latter enjoys the analytical tools of bi-orthogonal functions and determinantal point processes~\cite{Borodin1998}. Both aforementioned developments rely on the unitary (or orthogonal) bi-invariance of at least one of the two multiplied random matrices, meaning the ensemble is invariant under left- as well as right-multiplication of unitary (orthogonal) matrices. The considerations made in the present work exploits the same requirement.

Formerly, only products of complex Ginibre matrices~\cite{AB2012, AKW2013,KZ2014}, then of complex rectangular Gaussian matrices~\cite{AIK2013,IK2014,KS2014,ARRS2016,LW2016} and inverse Ginibre matrices~\cite{Forrester2014,ARRS2016}, and later of truncated unitary matrices (complex Jacobi ensemble)~\cite{ABKN2014,ARRS2016,KKS2016,LW2016} have been considered, see~\cite{AI2015} for a review on this development. These calculations have been possible due to known group integrals that are involved. Only a few years later the concept of harmonic analysis~\cite{Helgason2000} has been combined with these new developments to enlarge the class of non-classical ensembles to those of P\'olya ensembles~\cite{FKK2017,Kieburg2017,KFI2019} (originally known as polynomial ensembles of derivative type~\cite{KK2016,KK2019,KR2016}), see Definition~\ref{def:ensembles}, which also comprise some Muttalib-Borodin ensembles~\cite{Borodin1998,Muttalib1995}, for instance.

The name P\'olya ensemble has been dubbed to these ensembles due to their intimate relation to P\'olya frequency functions, that originated in approximation theory~\cite{Polya1913,Polya1915}. Because of the same importance of P\'olya frequency functions for sums of random matrices~\cite{KR2016,FKK2017}, we speak of multiplicative and additive P\'olya ensembles. These two kinds indeed differ by a subtle, but important feature which will not be specified here; we refer interested readers to~\cite{FKK2017}. Let us underline, throughout the present article we consider multiplicative P\'olya ensembles, only. Therefore, we omit the prefix ``multiplicative''.

P\'olya ensembles have two crucial advantages. Firstly, the multiplicative convolution is closed on their class so that they form a semi-group with the multiplication of a Haar distributed unitary matrix (also known as circular unitary matrices (CUE)) as the unit element. Secondly, the whole statistics essentially depends on a single univariate weight function. This fact simplifies the analysis drastically. For example, the harmonic analysis on matrix spaces reduces to its univariate counterpart. For products of complex random matrices this means that the spherical transform~\cite{Helgason2000,KK2016,KK2019} condenses to the Mellin transform in the one-dimensional case.

Very recently, products of real square matrices with real antisymmetric matrices and of quaternion matrices with anti-selfdual anti-Hermitian matrices have been studied~\cite{FILZ2019,KFI2019}. These works have followed an article on products of complex Ginibre matrices with a Gaussian unitary ensemble (GUE)~\cite{FIL2018}. Therein it has been shown that the natural group action of the matrix group $G$ on the corresponding Lie-algebra of its maximal compact subgroup might be indeed analytically feasible. To extend these results to more general ensembles, harmonic analysis seems to be again the ideal tool. Yet, for its successful application to the product of real asymmetric with real anti-symmetric matrices of even dimension a modification of the original theory by Harish-Chandra~\cite{Helgason2000} has been crucial~\cite{FILZ2019}. This can be all expected to hold true for the product of complex matrices with Hermitian matrices, as it is shown in the present work.

After introducing the necessary notation in Section~\ref{sec:pre}, we extend the concept of harmonic analysis to the set of Hermitian matrices with a fixed rank in Section~\ref{sec:spherical}. The spherical function and transform, introduced in this section, is here not anymore the one of the standard Fourier analysis, that corresponds to the additive convolution on Hermitian matrices. It reflects the multiplicative convolution and looks very similar to the original spherical function with an important difference, we need two copies of the complex plane for each of the ``Mellin-Fourier'' (frequency) parameter that encodes the sign of the eigenvalue. Furthermore we extend the harmonic analysis on the complex general linear group~\cite{Helgason2000,KK2016} to complex rectangular matrices. Quite naturally, orthogonal projections and inclusions play a particular outstanding role as has been already highlighted in~\cite{IK2014}.

Once the theoretical framework has been introduced, we consider the actual goal of the present article namely the product of complex rectangular and Hermitian random matrices. In Section~\ref{sec:mult.G}, we consider first the product of P\'olya ensembles of complex rectangular matrices with fixed Hermitian matrices. Therein, we compute the joint probability density, a corresponding set of bi-orthogonal functions, and an expression for the kernel. As a second example, we study the product of a P\'olya ensemble on the rectangular matrices with a polynomial ensemble drawn from the Hermitian matrices, in Section~\ref{sec:mult.H}. The change of the joint probability density, the bi-orthogonal functions, and the kernel will be the primary aim. Such transformation formulas have been considered in other works, too, that involve sums of random matrices~\cite{Kuijlaars2016,CKW2015,Kieburg2017} as well as products~\cite{KKS2016,KK2019,Kuijlaars2016,CKW2015}. Let us underline that all calculations are done at finite matrix dimensions, and that it is not the goal of the present work to investigate any limiting statistics. A technical work on a general product involving a GUE is currently in preparation~\cite{Kieburg2019}.

In Section~\ref{sec:conclusio}, we summarize and give a brief outlook on open problems. The details of the more extensive proofs are given in the Appendices.

\section{Preliminaries}\label{sec:pre}

In the ensuing sections, we make use of the particular matrix sets and $L^1$-function which we define here. For this purpose, we first of all introduce  the orthogonal projection/inclusion matrices of size $n\times m$,
\begin{equation}\label{proj-incl}
\Pi_{n,m}=\left\{\begin{array}{cl} \text{projection onto first } n \text{ rows}, & n<m, \\ \text{identity matrix } \leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_n, & n=m, \\ \text{inclusion into first } n \text{ rows}, & n>m. \end{array}\right.
\end{equation}
Moreover we denote the Hermitian adjoint of a matrix $g$ by $g^*$ and the Vandermonde determinant of an $n\times n$ diagonal matrix $a$ by $\Delta_n(a)=\prod_{n\geq c>d\geq 1}(a_c-a_d)$. All these definitions help in constructing the following sets:
\begin{enumerate}
\item		the complex general linear group of dimension $n\times n$: $G_{n}={\rm Gl}_{\mathbb{C}}(n)$ equipped with the flat Lebesgue measure $dg$,
\item		the $n\times n$ invertible Hermitian matrices $H_{n}$ equipped with the flat Lebesgue measure $dx$,
\item		the  $n\times n$ real diagonal and invertible $n\times n$ matrices: $D_n\simeq\mathbb{R}^n$ equipped with the flat Lebesque measure $da$,
\item		the  $n\times n$ positive real diagonal $n\times n$ matrices $A_n\simeq\mathbb{R}_+^n$ equipped with the flat Lebesque measure $da$,
\item		the group of the complex lower  $n\times n$ triangular matrices
 			$$T_n=\biggl\{t\in G\biggl| t=\{t_{ab}\}_{a,b=1\ldots,n}\ {\rm with}\ t_{ij}\in\mathbb{C},\ t_{ij}=0\ {\rm for}\ i<j\biggl\}$$
			equipped with the flat Lebesgue measure $dt$,
\item		the unitary group $K_n={\rm U}(n)$ equipped with the normalized Haar measure $d^*k$,
\item		the complex $l\times m$ matrices of rank $n\leq l,m$
			$$G_{l,m}^{(n)}=\{k\Pi_{l,n}g\Pi_{n,m}\tilde{k}^*|k\in K_l,\ \tilde{k}\in K_m,\ {\rm and}\ g\in G_n\},$$
			where $G_{n,n}^{(n)}=G_n$ and equipped with the measure $dg'=d(k\Pi_{l,n}g\Pi_{n,m}\tilde{k}^*)=dg d^*kd^*\tilde{k}$ induced by those on $G_n$, $K_l$ and $K_m$,
\item		the $l\times l$ Hermitan matrices of rank $n\leq l$
			$$H_l^{(n)}=\{k\Pi_{l,n}x\Pi_{n,l}k^*|k\in K_l\ {\rm and}\ x\in H_n\},$$
			where $H_n^{(n)}=H_n$ and equipped with the measure $dx'=d(k\Pi_{l,n}x\Pi_{n,l}k^*)=dx d^*k$ induced by those on $H_n$ and $K_l$,
\item		and the symmetric group $\mathbb{S}_n$ permuting $n$ elements.
\end{enumerate}
The matrices of lower rank than their matrix dimensions are of particular interest since they allow to deal with products of the form $g_1g_2$ with $g_1\in \mathbb{C}^{l,n}$, $g_2\in \mathbb{C}^{n,m}$ and $l,m>n$ as well as of the form $gxg^*$ with $g\in\mathbb{C}^{l,n}$, $x\in {\rm Herm}(n)$ and $l>n$. Obviously both products are of lower rank and can be considered as embeddings of lower dimensional matrices via multiplicative maps. We would like to underline that indeed all matrices of fixed lower rank can be represented as shown in the definition of the two sets $G_{l,m}^{(n)}$ and $H_l^{(n)}$. For example, for an $l\times l$ dimensional Hermitian matrix $x'$ of rank $n$,  one particular representation is by choosing for $x$ the non-zero eigenvalues of $x'$ on the diagonal and $k$ comprises the corresponding eigenvectors arranged as columns. The construction for $G_{l,m}^{(n)}$ is very similar.

Additionally, we need the set of $K$-invariant Lebesgue integrable functions on $G_{l,m}^{(n)}$ and $H_l^{(n)}$  and the symmetric integrable functions on $D_n$ and $A_n$. These are defined as
\begin{equation}\label{L1.def}
\begin{split}
L^{1,K}(G_{l,m}^{(n)})=&\{ f_G\in L^{1}(G_{l,m}^{(n)})|\ f_G(g)=f_G(k_1gk_2)\ {\rm for\ all}\ g\in G_{l,m}^{(n)},\ k_1\in K_l\ {\rm and}\ k_2\in K_m\},\\
L^{(1,K)}(H_l^{(n)})=&\{ f_H\in L^{1}(H_l^{(n)})|\ f_H(x)=f_H(kxk^*)\ {\rm for\ all}\ x\in H_l^{(n)}\ {\rm and}\ k\in K_l\},\\
L^{1,\mathbb{S}}(D_n)=&\{f_D\in L^{1}(D_n)|\ f_D(a)=f_D(\sigma a\sigma^T)\ {\rm for\ all}\ a\in D_n\ {\rm and}\ \sigma\in \mathbb{S}_n\},\\
L^{1,\mathbb{S}}(A_n)=&\{f_A\in L^{1}(A_n)|\ f_A(a)=f_A(\sigma a\sigma^T)\ {\rm for\ all}\ a\in A_n\ {\rm and}\ \sigma\in \mathbb{S}_n\}.
\end{split}
\end{equation}
The matrix $\sigma^T$ is the transpose of $\sigma$. We coin the subset of probability densities of these function spaces by $L_{\rm Prob}^{1,K}(G_{l,m}^{(n)})$, $L_{\rm Prob}^{1,K}(H_l^{(n)})$ $L_{\rm Prob}^{1,\mathbb{S}}(D_n)$ and $L_{\rm Prob}^{1,\mathbb{S}}(A_n)$ and adapt the notation of~\cite{KK2019}. Additionally we emphasise the space on which a function belongs by a subscript, like $f_G$, $f_H$, $f_D$ or $f_A$.

The spaces $L^{1,K}(G_{l,m}^{(n)})$ and $L^{1,\mathbb{S}}(A_n)$ as well as $L^{1,K}(H_l^{(n)})$ and $L^{1,\mathbb{S}}(D_n)$ are bijectively related via the isometrics
\begin{equation}\label{IGdef}
\mathcal{I}_G:\ L^{1,K}(G_{l,m}^{(n)})\rightarrow L^{1,\mathbb{S}}(A_n),\ f_A(a)=\mathcal{I}_Gf_G(a)=\frac{\pi^{n^2}}{n!}\left(\prod_{j=0}^{n-1}\frac{1}{(j!)^2}\right)\Delta_n^2(a)f_G(\Pi_{l,n}a\Pi_{n,m}).
\end{equation}
and
\begin{equation}\label{IHdef}
\mathcal{I}_H:\ L^{1,K}(H_l^{(n)})\rightarrow L^{1,\mathbb{S}}(D_n),\ f_D(a)=\mathcal{I}_Hf_H(a)=\frac{1}{n!}\left(\prod_{j=0}^{n-1}\frac{\pi^j}{j!}\right)\Delta_n^2(a)f_H(\Pi_{l,n}a\Pi_{n,l}).
\end{equation}
The bijectivity follows from the fact that the normalized Haar measure of the group $K_n$ is unique. Hence, we can go back and forth in the matrix spaces without losing any information.

\begin{remark}[Functions on the Complex Matrices $\mathbb{C}^{l\times m}$]\label{rem:complex}\

What might look strange is Eq.~\eqref{IGdef} since for the rectangular case $l\neq m$ we would expect an additional determinant. The reason why we do not have any here comes from the reference measure which is for a matrix $g'=k\Pi_{l,n} g\Pi_{n,m}\tilde{k}^* \in G_{l,m}^{(n)}$ with $g\in G_n$ given by the flat Lebesgue measure $dg$ (products of all real independent differentials) and $k\in K_l$ and $\tilde{k}\in K_m$ Haar distributed so that we have $dg'=dg d^*k d^*\tilde{k}$.

What is then the relation to $K$-invariant $L^1$-functions on $\mathbb{C}^{l\times m}$ that are distributed by the flat Lebesgue measure, since it is almost the set $G_{l,m}^{(n)}$ with $n=\min\{l,m\}$? First and foremost, all matrices of lower rank are of measure zero for this choice. Thus, we are compelled to look for another measure. In the case of a fully ranked matrix, say of rank $m$ with $l\geq m$, we choose a function $f_{\mathbb{C}}\in L^{1,K}(\mathbb{C}^{l\times m})$, in particular $f_{\mathbb{C}}(k\tilde{g}\tilde{k})=f_{\mathbb{C}}(\tilde{g})$ for all $\tilde{g}\in\mathbb{C}^{l\times m}$, $k\in K_{l}$ and $\tilde{k}\in K_m$. Then, the computation
\begin{equation}
\begin{split}
 \int_{\mathbb{C}^{l\times m}} d\tilde{g} f_{\mathbb{C}}(\tilde{g})=&\frac{\pi^{l m)}}{m!}\left(\prod_{j=0}^{m-1}\frac{1}{j!(j+l-m)!}\right)\int_{A_m}\Delta_m^2(a) \det a^{l-m} f_{\mathbb{C}}(\Pi_{l,m}a)\\
 =&\left(\prod_{j=0}^{m-1}\frac{\pi^{l-m}j!}{(j+l-m)!}\right)\int_{G_{l,m}^{(m)}}dg\int_{K_l}d^*k\int_{K_m}d^*\tilde{k} [\det (k\Pi_{l,m} g\tilde{k}^*)^*(k\Pi_{l,m} g\tilde{k}^*)]^{l-m}f_{\mathbb{C}}((k\Pi_{l,m} g\tilde{k}^*))
\end{split}
\end{equation}
leads to the identification of the corresponding $K$-invariant function on $G_{l,m}^{(m)}$ as
\begin{equation}\label{identification}
f_G(g')=\left(\prod_{j=0}^{m-1}\frac{\pi^{l-m}j!}{(j+l-m)!}\right)\det (g'{g'}^*)^{l-m}f_{\mathbb{C}}(g').
\end{equation}
Therefore, the well-known determinant is already part of $f_G$ and, hence, also comprised by $f_A$.
\end{remark}

The main focus of the present work is the multiplicative convolution
\begin{equation}\label{conv.H}
P_G\circledast Q_H(x)=\int_{G_{l,m}^{(n_1)}}d\tilde{g}\int_{H_m^{(n_2)}}d\tilde{x}\, \delta(x-\tilde{g}\tilde{x}\tilde{g}^*)\,P_G(\tilde{g})Q_H(\tilde{x})
\end{equation}
for $P_G$ an $L^1$-function on $G_{l,m}^{(n_1)}$ and $Q_H$ an  $L^1$-function on $H_m^{(n_2)}$, both being $K$-invariant. The Dirac delta function on $H_m^{(n_2)}$ is the one with respect to the measure $dx=d(k\Pi_{l,r}\hat{x}\Pi_{r,l} k^*)=d\hat{x} d^*k$ with $\hat{x}\in H_r$, $k\in K_l$ and $r=\min\{n_1,n_2\}$, i.e., $\int_{H_{l}^{(r)}}f(x)\delta(x-y)dx=f(y)$. It
can be explicitly evaluated for $l=m=n_1=n_2=n$,
\begin{equation}\label{conv.H-square}
P_G\circledast Q_H(x)=\int_{G_{n}}\frac{d\tilde{g}}{(\det \tilde{g}\tilde{g}^*)^{n}}P_G(\tilde{g})Q_H(\tilde{g}^{-1}x(\tilde{g}^{-1})^*).
\end{equation}
Its one dimensional counterpart is the well-known Mellin convolution
 \begin{equation}\label{uni-conv}
 p\circledast q(a)= \int_{0}^\infty\frac{da'}{a'}p(a')q(a/a')
 \end{equation}
 for $p\in L^1(\mathbb{R}_+)$ and $q\in L^1(\mathbb{R})$.

The convolution~\eqref{conv.H} is closely related to multiplicative convolutions on the complex rectangular matrices,
\begin{equation}\label{conv.G}
P_G\circledast Q_G(g)=\int_{G_{l,m}^{(n_1)}}d\tilde{g}_1\int_{G_{m,o}^{(n_2)}}d\tilde{g}_2\,\delta(g-\tilde{g}_1\tilde{g}_2)\,P_G(\tilde{g}_1)Q_G(\tilde{g}_2)
\end{equation}
for two $K$-invariant $L^{1}$-functions on $G_{l,m}^{(n_1)}$ and $G_{m,o}^{(n_2)}$, respectively. Again, the Dirac delta function is the one with respect to the measure $dg=d(k\Pi_{l,r}\hat{g}\Pi_{r,o} \tilde{k}^*)=d\hat{g}d^*kd^*\tilde{k}$ with $\hat{g}\in G_r$, $k\in K_l$, $\tilde{k}\in K_o$ and the rank $r=\min\{n_1,n_2\}$.
For the group $G_n$, which is the case $l=m=o=n_1=n_2=n$, this evaluates to the well-known convolution
\begin{equation}\label{conv.G-square}
P_G\circledast Q_G(g)=\int_{G_n}\frac{d\tilde{g}}{(\det \tilde{g}\tilde{g}^*)^n}P_G(\tilde{g})Q_G(g\tilde{g}^{-1}).
\end{equation}
The latter convolution has been studied intensively in the literature~\cite{Helgason2000}. In random matrix theory, Eq.~\eqref{conv.G} has recently excited interests in applications in wireless telecommunications (e.g., see~\cite{TV2004,AKW2013,WZCT2015,KAAC2019}), quantum information~\cite{RSZ2011,Lakshminarayan2013,CHN2017}, and machine learning~\cite{HN2018,LC2018,TWJTN2019}. This culminated in rapid developments~\cite{AI2015} and has been connected to harmonic analysis~\cite{KK2016,KK2019,FKK2017}. Exactly the harmonic analysis approach is the path will pursue here, too.

To this aim, we need the Mellin transform on the full real line and not only on the half line. It can be considered as the direct sum of the Mellin transform of functions on the positive and negative real line, i.e.,
\begin{equation}\label{Mellin}
\begin{split}
\mathcal{M}: L^1(\mathbb{R})\rightarrow \mathcal{M} L^1(\mathbb{R});\qquad f\mapsto\mathcal{M}f(s,L)=\int_\mathbb{R}\frac{dx}{|x|}[{\rm sign\,}(x)]^L |x|^{s}f(x)
\end{split}
\end{equation}
for any $s\in\mathbb{C}$ and $L\in\{0,1\}$ where the integrand is Lebesgue integrable.  The function ${\rm sign}(x)$ is the signum function which yields the sign of a real number $x$ and vanishes if $x=0$.
Indeed, any function $f\in L^1(\mathbb{R})$ can be decomposed into $f(x)=f_+(x)+f_-(-x)$ with $f_{\pm}(\pm x)=f(x)\Theta(\pm x)$ where $\Theta$ is the Heaviside step-function. Then the ordinary Mellin transform is related to the one on the real line as $\mathcal{M}f(s,L)=\mathcal{M}f_+(s)+(-1)^L\mathcal{M}f_-(s)$. Due to the two copies of the complex plane $s\in\mathbb{C}$ denoted by $L\in\mathbb{Z}_2$, the Mellin transform~\eqref{Mellin} is still invertible and has the inverse
\begin{equation}\label{inv.Mellin}
\mathcal{M}^{-1}[\mathcal{M}f](x)=\lim_{\epsilon\to0}\sum_{L=0,1}\frac{[{\rm sign\,}(x)]^L}{4\pi}\int_{-\infty}^\infty ds \mathcal{M}f(\imath s+1,L)|x|^{-\imath s-1}\zeta_1(\epsilon s).
\end{equation}
The regularization
\begin{equation}
\zeta_1(\epsilon s)=\frac{\pi^2\cos(\epsilon s)}{\pi^2-4\epsilon^2 s^2}
\end{equation}
 guarantees that the  inverse holds for any $L^1$-function on the real line, and it can be omitted when the other terms in the integral~\eqref{inv.Mellin} are absolutely integrable. The sum over $L$ properly combines the two pieces $f_+$ and $f_-$ of a function $f$.

 The univariate Mellin transform decouples the multiplicative univariate convolution~\eqref{uni-conv}, i.e.,
 \begin{equation}\label{Mellin.fact}
 \mathcal{M}(p\circledast q)(s,L)=\mathcal{M} p(s) \mathcal{M} q(s,L)
 \end{equation}
 for $p\in L^1(\mathbb{R}_+)$ and $q\in L^1(\mathbb{R})$. We will construct the counterpart in matrix space.

In the next section, we generalize the construction~\eqref{Mellin} to the multivariate case of Hermitian matrices.

\section{Spherical Transforms}\label{sec:spherical}

The theory of spherical transforms~\cite{Helgason2000} developed by Harish-Chandra et al. in the 50's has been extremely helpful in dealing with sums~\cite{KR2016,FKK2017} and products~\cite{KK2016,KK2019,KFI2019} of random matrices. However, already when dealing with products of real antisymmetric and real asymmetric matrices, see~\cite{KFI2019}, we have seen that the original definition of the spherical transform does not carry far and needs to be modified. This is also here the case. The first step to achieve this goal is the definition of the spherical functions, which serve as the Fourier factors in ordinary Fourier analysis. To this goal, we define the two diagonal matrices
\begin{equation}\label{def.L.s}
s^{(n)}={\rm diag\,}(n-1,n-2,\ldots,2,1,0)\in\mathbb{C}^n\qquad{\rm and}\qquad L^{(n)}={\rm diag\,}({\rm mod}_2(n-1),{\rm mod}_2(n-2),\ldots,0,1,0)\in\mathbb{Z}_2^n.
\end{equation}

\begin{definition}[Spherical Functions on $H_{l}^{(n)}$ and $G_{l,m}^{(n)}$]\label{def:spherical-functions}\

Let $x\in H_{l}^{(n)}$ and $g\in G_{l,m}^{(n)}$ be two fixed matrices with the three positive integers $n\leq l,m$. Moreover, we set $s_{n+1}=-1$ and $L_{n+1}=-1$ and specify the rectangular matrix $\Pi_{j,l}$ as in Eq.~\eqref{proj-incl}. We define
\begin{enumerate}
\item		the spherical function on $H_{l}^{(n)}$ by
			\begin{equation}\label{spher:as}
				\Phi(s,L; x)=\frac{\int_{K_l}d^*k\prod_{j=1}^{n} {\rm sign}[\det \Pi_{j,l}kxk^*\Pi_{l,j}]^{L_j-L_{j+1}-1} |\det \Pi_{j,l}kxk^*\Pi_{l,j}|^{s_j-s_{j+1}-1}}{\int_{K_l}d^*k\prod_{j=1}^{n} |\det \Pi_{j,l}k\Pi_{l,n}\Pi_{n,l}k^*\Pi_{l,j}|^{s_j-s_{j+1}-1}}
			\end{equation}
			for $s={\rm diag\,}(s_1,\ldots,s_n)\in\mathbb{C}^n$ and $L={\rm diag\,}(L_1,\ldots,L_n)\in\mathbb{Z}_2^n$ with ${\rm Re}\,(s_j-s_{j+1})\geq 1$ for all $j=1,\ldots,n$ and analytically continue $\Phi$ to ${\rm Re}\,(s_b-s_{b+1})< 1$ for some $b=1,\ldots, n$,
\item		and the spherical function on $G_{l,m}^{(n)}$, cf., Ref.~\cite{Helgason2000,KK2016} for $G_n$,
			\begin{equation}\label{spher:s}
				\Psi(s; g)=\Phi(s,L^{(n)}; gg^*)
			\end{equation}
			for all $s\in\mathbb{C}^n$ and fixed $L^{(n)}$ as in Eq.~\eqref{def.L.s}.
\end{enumerate}
\end{definition}

We would like to underline that we employ here a different convention of $s_{n+1}$ compared to the standard literature~\cite{Helgason} where its value is usually $-(n+1)/2$. We have decided for the choice $s_{n+1}=-1$ so that the notation becomes simpler. One particular consequence is that the choice of $s$ to find the normalization of the spherical functions, which is $\Phi(s^{(n)},L^{(n)}; x)=1$ and $\Psi(s^{(r)}; g)=1$. Moreover, we have the trivial normalizations $\Phi(s,L; \Pi_{l,n}\Pi_{n,l})=1$ and $\Psi(s; \Pi_{l,n}\Pi_{n,m})=1$; the latter is due to $\Pi_{n,m}\Pi_{m,n}=\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_n$. The normalising denominator in~\eqref{spher:as}, which we denote by
\begin{equation}\label{normalization}
C_{l,n}(s)=\int_{K_l}d^*k\prod_{j=1}^{n}|\det \Pi_{j,l}k\Pi_{l,n}\Pi_{n,l}k^*\Pi_{l,j}|^{s_j-s_{j+1}-1},
\end{equation}
is unity in the case of maximal rank, i.e., $l=n$.

\begin{remark}[Limit to Lower Ranked Matrices]\label{rem:limit}\

Evidently, the spherical functions $\Psi$ and $\Phi$ are $K$-invariant so that we can also study these function on $A_n$ and $D_n$, respectively. The relation between the spherical function on $G_{l,m}^{(n)}$ to the one on $G_{l,m}^{(n-1)}$ is given as follows
\begin{equation}
\begin{split}
&\lim_{a_n\to0} \lim_{s_n\to 0}C_{m}^{(n)}({\rm diag\,}(s_1,\ldots,s_n))\Psi({\rm diag\,}(s_1,\ldots,s_n); {\rm diag\,}(a_1,\ldots,a_n))\\
=&C_{m}^{(n-1)}({\rm diag\,}(s_1,\ldots,s_{n-1}))\Psi({\rm diag\,}(s_1,\ldots,s_{n-1})-\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{n-1}; {\rm diag\,}(a_1,\ldots,a_{n-1})).
\end{split}
\end{equation}
The order of the limits is of paramount importance otherwise it is zero or infinity, depending on the exponent, due to the vanishing determinant $\det a$ in the definition~\eqref{spher:as}. Similarly for the Hermitian matrices we find
\begin{equation}
\begin{split}
&\lim_{a_n\to0} \lim_{s_n, L_n\to 0}C_{m}^{(n)}({\rm diag\,}(s_1,\ldots,s_n))\Phi({\rm diag\,}(s_1,\ldots,s_n),{\rm diag\,}(L_1,\ldots,L_n); {\rm diag\,}(a_1,\ldots,a_n))\\
=&C_{m}^{(n-1)}({\rm diag\,}(s_1,\ldots,s_{n-1}))\Phi({\rm diag\,}(s_1,\ldots,s_{n-1})-\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{n-1},{\rm diag\,}(L_1,\ldots,L_{n-1})-\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{n-1}; {\rm diag\,}(a_1,\ldots,a_{n-1})).
\end{split}
\end{equation}
Relations with even lower ranks can be computed recursively. It is always crucial to multiply the normalization to the spherical function to find a smooth transition from higher to lower ranked matrices.
\end{remark}

The spherical function $\Psi$ for $G_n$ has a remarkable explicit analytical representation in terms of the squared singular values $a\in A_n$ of $g\in G_n$ given by the Gelfand-Na\u{\i}mark integral~\cite{GN1957}
\begin{equation}\label{GN-integral}
\Psi(s; g)=\Phi(s,L^{(n)}; a)=\left(\prod_{j=0}^{n-1}j!\right)\frac{\det[a_c^{s_b}]_{b,c=1,\ldots,n}}{\Delta_n(a)\Delta_n(s)}.
\end{equation}
The corresponding formula for general rectangular matrices $g\in G_{l,m}^{(n)}$ and Hermitian matrices $x\in H_{l}^{(n)}$ is very similar.

\begin{theorem}[Spherical Functions $\Phi$ and $\Psi$]\label{thm:spher.func}\

\begin{enumerate}
\item		Let $a\in D_n$ and $s\in\mathbb{C}^n$ with non-degenerate spectra, i.e. $a_l\neq a_k$ and $s_l\neq s_k$ for all $l\neq k$,  and $L\in\mathbb{Z}_2^n$. The spherical function~\eqref{spher:as} has the explicit form
			\begin{equation}\label{spher:as.b}
				\Phi(s,L; a)=\left(\prod_{j=0}^{n-1}j!\right)\frac{\det[[{\rm sign}(a_c)]^{L_b}|a_c|^{s_b}]_{b,c=1,\ldots,n}}{\Delta_n(a)\Delta_n(s)}.
			\end{equation}
\item			Let $a\in A_n$ and $s\in\mathbb{C}^n$ with non-degenerate spectra. The spherical function~\eqref{spher:s} is
			\begin{equation}\label{spher:s.b}
				\Psi(s; a)=\left(\prod_{j=0}^{n-1}j!\right)\frac{\det[a_c^{s_b}]_{b,c=1,\ldots,n}}{\Delta_n(a)\Delta_n(s)}.
			\end{equation}
\item		The normalization~\eqref{normalization} is equal to
			\begin{equation}\label{normalization.b}
				C_{l,n}(s)=\prod_{j=1}^n\frac{(l-j)!\Gamma[s_j+1]}{(n-j)!\Gamma[s_j+l-n+1]}
			\end{equation}
			for any $s\in\mathbb{C}^{n}$.
\end{enumerate}

\end{theorem}

This theorem is proven in Appendix~\ref{Proof:spher.func} in a very similar way as the real counterpart with even dimensional antisymmetric matrices in~\cite{KFI2019}. This theorem shows that we can deal with all products of the form $gxg^*$ with $x$ being $m\times m$ Hermitian and $g$ a complex $n\times m$ rectangular with $m\leq n$ in a unified way. The following Proposition~\ref{prop:fact.spher} highlights this. Moreover, the case $m>n$ in the above product  can be considered, too, which is essentially a projection.

In the next step, we need to show a factorization theorem for the spherical transforms to be applicable to the convolution~\eqref{conv.H}.  For the spherical function~\eqref{spher:s} of the group $G_n$, this particular factorization reads~\cite{Helgason2000,KK2016}
\begin{equation}\label{factorization.square}
\int_{K_{n}}d^* k \Psi(s; gkg')=\Psi(s; g)\,\Psi(s;g')
\end{equation}
for any two $g,g'\in G_n$. Something similar is also true for the spherical function~\eqref{spher:as} as well as for the rectangular case which is proven in Appendix~\ref{Proof:fact.spher}.

\begin{proposition}[Factorization of $\Phi$ and $\Psi$]\label{prop:fact.spher}\

\begin{enumerate}
\item		Let $g\in G_{l,m}^{(n_1)}$ and $x\in H_m^{(n_2)}$ with $r=\min\{n_1,n_2\}$ the rank of $gxg^*$. Additionally, we choose $\tilde{s}\in\mathbb{C}^r$ and $\tilde{L}\in\{0,1\}^r$. Then, we find the following factorization
\begin{equation}\label{factorization}
\begin{split}
\int_{K_m}d^*k \Phi(s,L; gkxk^*g^*)=C_{m,r+|n_1-n_2|}(\tilde{s})\left\{\begin{array}{cl} \Psi(s; g)\ \Phi(\tilde{s},\tilde{L}; x), & n_1=r,\\ \Psi(\tilde{s}; g)\Phi(s,L; x), & n_2=r \end{array}\right.
\end{split}
\end{equation}
for any $s\in\mathbb{C}^r$ and $L\in\mathbb{Z}_2^r$. We defined $\tilde{s}={\rm diag\,}(s+|n_1-n_2|\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{r},s^{(|n_1-n_2|)})$ and $\tilde{L}={\rm diag\,}(L+|n_1-n_2)\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{r},L^{(|n_1-n_2|)})$.
\item		Let $g_1\in G_{l,m}^{(n_1)}$ and $g_2\in G_{m,o}^{(n_2)}$ with $r=\min\{n_1,n_2\}$ the rank of $g_1g_2$. Then, the factorization formula reads
\begin{equation}\label{factorization.rect}
\begin{split}
\int_{K_m}d^*k \Psi(s,L; g_1kg_2)=C_{m,r+|n_1-n_2|}(\tilde{s})\left\{\begin{array}{cl} \Psi(s; g_1)\Psi(\tilde{s}; g_2), & n_1=r,\\ \Psi(\tilde{s}; g_1)\Psi(s; g_2), & n_2=r \end{array}\right.
\end{split}
\end{equation}
for all $s\in\mathbb{C}^r$, where $\tilde{s}$  is defined as before.
\end{enumerate}

\end{proposition}

The second part of Proposition~\ref{prop:fact.spher} is equivalent to the discussion in~\cite{IK2014} for the product of complex rectangular random matrices stating that the singular value statistics is the same as product of a product of square random matrices at the expense of a random projection and the multiplication with a power of a determinant. The projection is represented here by the constants on the right hand side of Eq.~\eqref{factorization.rect}, which are essentially spherical transforms of random projections, see below.  The additional determinant in the weight is reflected by the shift of $s$ in $\Psi$. The latter can be seen by the identity
\begin{equation}\label{shiftPsi}
\Psi(s+\mu\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_n;g)=(\det gg^*)^\mu\Psi(s;g),
\end{equation}
for $g\in G_n=G_{n,n}^{(n)}$. A similar identity exists for the spherical transform on the Hermitian matrices $H_n=H_n^{(n)}$,
\begin{equation}\label{shiftPhi}
\Phi_{n}(s+\mu\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_n,L+j\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_n;x)=({\rm sign\,}(\det x))^j |\det x|^\mu \Phi(s,L;x)
\end{equation}
for any $\mu\in\mathbb{C}$ and $j\in\mathbb{Z}$.

\begin{remark}[Relation between Spherical Functions of $g$ and $g^*$]\label{rem:rel.adj}\

Another thing, which is worth mentioning, is that $\Psi(s; g)=\Phi(s,L^{(n)}; gg^*)$ only depends on the singular values $a\in A_n$ of the matrix $g$ which are shared by its Hermitian adjoint $g^*$. Thence, one may ask what the relation between both of their spherical functions is. The answer follows from Proposition~\ref{prop:fact.spher} above in combination with Definition~\ref{def:spherical-functions}. Without restriction of generality we assume $l\geq m$, then we calculate
\begin{equation}\label{spher-rel}
\begin{split}
\Psi(s; g)=\Phi(s,L^{(n)}; gg^*)=&\Phi(s,L^{(n)}; \Pi_{l,n}a\Pi_{n,l})\\
=&\int_{K_m}d^*k\Phi(s,L^{(n)}; \Pi_{l,m}k\Pi_{m,n}a\Pi_{n,m}k^*\Pi_{m,l})\\
=&C_{m,m}(\tilde{s})\Psi(\tilde{s}; \Pi_{l,m}) \Phi(s,L^{(n)}; \Pi_{m,n}a\Pi_{n,m})=\Phi(s,L^{(n)}; g^*g)=\Psi(s; g^*)
\end{split}
\end{equation}
with $\tilde{s}={\rm diag\,}(s+(m-n)\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{n},s^{(m-n)})$.
In the first and second line, we have exploited the invariance of the Haar measure under multiplying unitary matrices, especially we have introduced the unitary matrix ${\rm diag\,}(k,\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{l-m})\in K_l$ which commutes with the projector $\Pi_{l,m}$ like ${\rm diag\,}(k,1_{l-m})\Pi_{l,m}=\Pi_{l,m}k$ and can be Haar distributed, too. In the third line we employed $C_{m,m}(\tilde{s})=1$. The spherical function $\Psi$ drops out because it is normalized for the projection $\Pi_{l,m}$ which has maximal rank.

This result shows that the spherical transform is the same for $g$ and $g^*$. This is not immediately clear from the definition because we integrate over different groups. Indeed without the normalization in the denominator~\eqref{spher:as}, we would have found a difference between both spherical transforms.
\end{remark}

The full benefit of Propositions~\ref{prop:fact.spher} unfurls when combining the factorization with the spherical transform, which is the Fourier transform on curved symmetric spaces. They are defined as follows.

\begin{definition}[Spherical Transforms corresponding to $\Phi_n$ and $\Psi_n$]\label{def:spher.trans}\

 Let $Q_G\in L^{1,K}(G_{l,m}^{(n)})$,  $P_H\in L^{1,K}(H_m^{(n)})$, $q_A=\mathcal{I}_GQ_G$, and $p_D=\mathcal{I}_HP_H$, see Eqs.~\eqref{IGdef} and~\eqref{IHdef}.
\begin{enumerate}
\item		The spherical transform $\mathcal{S}_{\Phi}: L^{1,K}(H_m^{(n)})\rightarrow \mathcal{S}_{\Phi}(L^{1,K}(H_m^{(n)}))$ corresponding to $\Phi$ is defined as
			\begin{equation}\label{Strafo:as}
				\begin{split}
				\mathcal{S}_{\Phi}P_H(s,L)=&\int_{H_m^{(n)}} dx P_H(x)\Phi(s,L; x)\\
				=&\left(\prod_{j=0}^{n-1}j!\right)\int_{D_n}da\, p_D(a)\frac{\det[[{\rm sign}(a_c)]^{L_b}|a_c|^{s_b}]_{b,c=1,\ldots,n}}{\Delta_n(a)\Delta_n(s)}=\mathcal{S}_{\Phi}p_D(s,L)
				\end{split}
			\end{equation}
			for those $s\in\mathbb{C}^n$ for which the integral exists and for any $L\in\mathbb{Z}_2^n$.
\item		The spherical transform $\mathcal{S}_{\Psi}: L^{1,K}(G_{l,m}^{(n)})\rightarrow \mathcal{S}_{\Psi}(L^{1,K}(G_{l,,m}^{(n)}))$ corresponding to $\Psi$ is (see~\cite{Helgason2000,KK2016} for $G_n$)
			\begin{equation}\label{Strafo:s}
			\begin{split}
			\mathcal{S}_{\Psi}Q_G(s)=&\int_{G_{l,m}^{(n)}}dg Q_G(g)\Psi_n(s; g)=\left(\prod_{j=0}^{n-1}j!\right)\int_{A_n}da\, q_A(a)\frac{\det[a_c^{s_b}]_{b,c=1,\ldots,n}}{\Delta_n(a)\Delta_n(s)}=\mathcal{S}_{\Psi}q_A(s)
			\end{split}
			\end{equation}
			for any $s\in\mathbb{C}^n$ where the integral exists.
\end{enumerate}
In the second equalities, we slightly abuse notation and have to assume that $s_l\neq s_k$ for $l\neq k$.
\end{definition}

The spherical transforms are normalized such that
\begin{equation}
\mathcal{S}_{\Phi}P_H(s^{(n)},L^{(n)})=\int_{H_m^{(n)}} dx\,P_H(x)\qquad{\rm and}\qquad
\mathcal{S}_{\Psi}Q_G(s^{(n)})=\int_{G_{l,m}^{(n)}} dg\,Q_G(g).
\end{equation}
These relations are extremely helpful when fixing the constants for explicit ensembles.

\begin{example}[Spherical Transform of a Sub-Block of a Unitary Matrix]\label{ex:uni}\

As a simple example we would like to compute the spherical transform of a $l\times m$ sub-block matrix $g=\Pi_{l,M}k\Pi_{M,m}$ of a Haar distributed unitary matrix $k\in K_M$ with $l,m\leq M$. The probability density is then given by $P_G(g)=\int_{K_M}d^*k\delta(g-\Pi_{l,M}k\Pi_{M,m})$ with the Dirac delta function in the space $G_{l,m}^{(n)}$ with $n=\min\{l,m\}$. Without restriction of generality due to Eq.~\eqref{spher-rel}, we assume that $l\leq m$. Then we have
\begin{equation}\label{spher.uni}
\begin{split}
\int_{K_M} d^*k \Psi(s;\Pi_{l,M}k\Pi_{M,m})=C_{M,m}({\rm diag\,}(s+(m-l)\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_l,s^{(m-l)}))=\prod_{j=1}^l\frac{(M-j)!\Gamma[s_j+m-l+1]}{(m-j)!\Gamma[s_j+M-l+1]}
\end{split}
\end{equation}
because of Eq.~\eqref{factorization.rect} and the normalization of the spherical functions for orthogonal projections. Let us underline that we have not needed any restriction of $l$ and $m$ with respect to $M$ as it is has been the case in~\cite{KKS2016}. This is one of the strengths of the harmonic analysis approach presented here.
\end{example}

A direct consequence of Definition~\ref{def:spher.trans} in combination with Proposition~\ref{prop:fact.spher} is the factorization of the spherical transform when employing it on the convolutions~\eqref{conv.H} and~\eqref{conv.G}. For the convolution~\eqref{conv.G-square} on the square complex matrices $G_{n}$  this is well-known~\cite{Helgason2000,KK2016,KK2019} and explicitly reads
\begin{equation}
\mathcal{S}_{\Psi}[P_G\circledast Q_G](s)=\mathcal{S}_{\Psi}P_G(s)\mathcal{S}_{\Psi}Q_G(s)
\end{equation}
for two functions $P_G,Q_G\in L^{1,K}(G_{n})$.
The following corollary of Definition~\ref{def:spher.trans} and Proposition~\ref{prop:fact.spher} extends this result.

\begin{corollary}[Factorization Formulas of $S_\Phi$ and $S_\Psi$]\label{cor:fact:trans}\

\begin{enumerate}
\item		Let $P_G\in L^{1,K}(G_{l,m}^{(n_1)})$ and $Q_H\in L^{1,K}(H_m^{(n_2)})$ with $r=\min\{n_1,n_2\}$. Then the spherical transform of the convolution $P_G\circledast Q_H$ is
\begin{equation}\label{factorization.conv}
\begin{split}
\mathcal{S}_\Phi[P_G\circledast Q_H](s,L)=C_{m,r+|n_1-n_2|}(\tilde{s})\left\{\begin{array}{cl} \mathcal{S}_\Psi P_G(s)\mathcal{S}_\Phi Q_H(\tilde{s},\tilde{L}), & n_1=r,\\  \mathcal{S}_\Psi P_G(\tilde{s})\mathcal{S}_\Phi P_H(s,L), & n_2=r,\end{array}\right.
\end{split}
\end{equation}
			with $\tilde{s}={\rm diag\,}(s+|n_1-n_2|\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{r},s^{(|n_1-n_2|)})$ and $\tilde{L}={\rm diag\,}(L+|n_1-n_2|\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{r},L^{(|n_1-n_2|)})$.
\item		For $P_G\in L^{1,K}(G_{l,m}^{(n_1)})$ and $Q_G\in L^{1,K}(G_{m,o}^{(n_2)})$ with $r=\min\{n_1,n_2\}$, the spherical transform of the convolution $P_G\circledast Q_G$ becomes
\begin{equation}\label{factorization.rect.conv}
\begin{split}
\mathcal{S}_\Psi[P_G\circledast Q_G](s)=C_{m,r+|n_1-n_2|}(\tilde{s})\left\{\begin{array}{cl} \mathcal{S}_\Psi P_G(s)\mathcal{S}_\Psi Q_G(\tilde{s}), & n_1=r,\\ \mathcal{S}_\Psi P_G(\tilde{s})\mathcal{S}_\Psi Q_G(s), & n_2=r, \end{array}\right.
\end{split}
\end{equation}
			where $\tilde{s}={\rm diag\,}(s+|n_1-n_2|\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{r},s^{(|n_1-n_2|)})$.
\end{enumerate}
\end{corollary}

\begin{proof}
The second part of this  corollary about the spherical transform on the complex matrices immediately results from the first due to the relation of the two spherical functions $\Phi$ and $\Psi$. Therefore, we only concentrate on the proof of the first part on the Hermitian matrices.

We consider the spherical transform
\begin{equation}\label{proof:fact:trans.1}
\begin{split}
\mathcal{S}_{\Phi}[P_G\circledast Q_H](s,L)=\int_{G_{l,m}^{(n_1)}}d g\int_{H_m^{(n_2)}} dx\Phi(s,L;gxg^*)P_G(g)Q_H(x),
\end{split}
\end{equation}
where we have evaluated the Dirac delta function in Eq.~\eqref{conv.H}. Now we use the fact that $P_G$ is $K$-invariant and introduce a Haar distributed unitary matrix $k\in K_m$ in the following way $g\to gk$. For the integration over $k$ we employ Proposition~\ref{prop:fact.spher} and find the claim~\eqref{factorization.conv}.
\end{proof}

As a last ingredient for solving the convolution~\eqref{conv.H} we need the invertibility of  the spehrical transform $S_\Phi$. When we have the inverse, the convolution can be rewritten as follows
\begin{equation}
P_G\circledast Q_H=\mathcal{S}_\Phi^{-1}[\mathcal{S}_{\Psi}P_G\,\mathcal{S}_{\Phi}Q_H].
\end{equation}
As we know from univariate probability theory, this representation is advantageous because the probabilistic averages can be readily carried out. We have proved the invertibility of the spherical transform in Appendix~\ref{Proof:inv} and the proper statement reads:

\begin{proposition}[Inverse of $S_\Phi$ and $S_\Psi$]\label{prop:inv}\

We define define the auxiliary function
\begin{equation}\label{xi.def}
\zeta_n(z)=\frac{\cos(z)}{\prod_{k=1}^n[1-4z^2/(\pi(2k-1))^2]}.
\end{equation}
\begin{enumerate}
\item		Let $P_H\in L^{1,K}(H_m^{(n)})$ and $p_D=\mathcal{I}_HP_H\in L^{1,\mathbb{S}}(D_n)$.
			The spherical transform $\mathcal{S}_{\Phi}$ is injective and, hence, invertible when restricted to its image. The inverse has the explicit form
			\begin{equation}\label{Strafo:as:inv}
			\begin{split}
			p_D(a)=\mathcal{S}_{\Phi}^{-1}[\mathcal{S}_{\Phi}p_D](a)=&\sum_{L\in\{0,1\}^n}\frac{\Delta_n(a)}{(n!)^2 \prod_{j=0}^{n-1}j!}\lim_{\epsilon\to0}\int_{\mathbb{R}^n} \frac{ds}{(4\pi)^{n}}\mathcal{S}_{\Phi}p_D(\imath s+s^{(n)},L)\prod_{l=1}^n\zeta_n(\epsilon s_l)\\
			&\times\Delta_n(\imath s+s^{(n)})\det[[{\rm sign}(a_c)]^{L_b}|a_c|^{-\imath s_b-n+b-1}]_{b,c=1,\ldots,n}
			\end{split}
			\end{equation}
			for almost all $a\in D_n$.
\item		Choosing $P_G\in L^{1,K}(G_{l,m}^{(n)})$ and $p_A=\mathcal{I}_GP_H\in L^{1,\mathbb{S}}(A_n)$, the spherical transform $\mathcal{S}_{\Psi}$ is invertible when restricted to its image with the inverse
			\begin{equation}\label{Strafo:s:inv}
			\begin{split}
			p_A(a)=\mathcal{S}_{\Psi}^{-1}[\mathcal{S}_{\Psi}p_A](a)=&\frac{\Delta_n(a)}{(n!)^2 \prod_{j=0}^{n-1}j!}\lim_{\epsilon\to0}\int_{\mathbb{R}^n} \frac{ds}{(2\pi)^{n}}\mathcal{S}_{\Psi}p_A(\imath s+s^{(n)})\prod_{l=1}^n\zeta_n(\epsilon s_l)\\
			&\times\Delta_n(\imath s+s^{(n)})\det[a_c^{-\imath s_b-n+b-1}]_{b,c=1,\ldots,n}
			\end{split}
			\end{equation}
			for almost all $a\in A_n$. For $G_n$ this statement is equal to the one in~\cite{KK2016}.
\end{enumerate}
\end{proposition}

As in the univariate case, the regularization $\zeta_n(z)$ is only needed when the the spherical transform is not $L^1$-integrable

Now, we have developed the theoretical framework to deal with products of random matrices on the Hermitian matrices. But before we come to that we want to underline that the above discussion has been true also for the whole set of $L^1$-functions which are $K$-invariant and not only for probability densities. In the ensuing discussion, when studying random matrices we restrict ourselves to the latter.

\section{Multiplicative Convolution on $G={\rm Gl}_{\mathbb{C}}(n)$ and $H={\rm Herm}(n)$}\label{sec:conv}

Already in previous works~\cite{KK2016,KK2019,KR2016,FKK2017,Kieburg2017,KFI2019}, it has been pointed out that particular ensembles are analytically easier to handle than others, in particular the integrals of the spherical transform and its inverse can be explicitly carried out. Here, one class of ensembles, called the polynomial ensembles~\cite{KZ2014}, are preferable to deal with since they always exhibit a determinantal point process even at finite matrix dimension~\cite{Borodin1998}. However, the convolution of two polynomials ensembles do not usually yield a polynomial ensembles. To overcome this obstacle a subclass has been identified~\cite{KK2016,KK2019,KR2016,FKK2017}. This subclass has been first coined the polynomial ensembles of derivative type~\cite{KK2016,KK2019,KR2016} due to its special form but then renamed to P\'olya ensembles~\cite{FKK2017}. The reason for the re-baptism is the fact that the ensembles are bijectively related to P\'olya frequency functions~\cite{Polya1913,Polya1915} that satisfy some differentiability and integrability conditions. The definition of these two sets requires two sets of $L^{1}$ functions on a subset $\mathbb{I}\subset\mathbb{R}$,
\begin{equation}\label{Ln1}
\begin{split}
L_n^{1}(\mathbb{I})=&\left\{f\in L_n^{1}(\mathbb{I})\left| \int_{\mathbb{I}}dx |x^j f(x)|<\infty\ {\rm for\ all}\ j\in[0,n]\right\}\right.
\end{split}
\end{equation}
and
\begin{equation}\label{Pn1}
\begin{split}
\mathcal{P}_n^{1}=&\left\{f\in L_n^{1}(\mathbb{R}_+)| f\circ\exp\ {\rm is}\ n\text{-times\ differentiable\ and\ a P\'olya\ frequency\ function\ of\ order}\ n, {\rm and}\right.\\
&\left. (x\partial)^jf\in L_n^{1}(\mathbb{R}_+)\  {\rm for\ all}\ j=0,\ldots,n\right\},
\end{split}
\end{equation}
where $f\circ\exp: \mathbb{R}\to\mathbb{R}$  is given by $f\circ\exp(x)=f(e^x)$.
We recall that a P\'olya frequency function is given as a real, non-zero function on $\mathbb{R}$ that satisfies the inequality~\cite{Polya1913,Polya1915}
\begin{equation}
\Delta_j(x)\Delta_j(y)\det[f(x_b-y_c)]_{b,c=1,\ldots,j}\geq0
\end{equation}
for all $x,y\in \mathbb{R}^j$ and $j=1,\ldots,n$.

\begin{definition}[Polynomial and P\'olya Ensembles~\cite{KZ2014,KK2016,KR2016,FKK2017}]\label{def:ensembles}\

\begin{enumerate}
\item		A polynomial ensemble on $G_{l,m}^{(n)}$ associated to the weights $w_1,\ldots,w_n\in L_{n-1}^1(\mathbb{R}_+)$ is a $K$-invariant ensemble whose squared singular values $a\in A$ are distributed as
			\begin{equation}\label{pol.G}
			p_A(a)=\frac{1}{n!}\Delta_n(a)\frac{\det[w_b(a_c)]_{b,c=1,\ldots,n}}{\det[\mathcal{M}w_b(c)]_{b,c=1,\ldots,n}}\in L_{\rm Prob}^{1}(A_n).
			\end{equation}
\item		A P\'olya ensemble on $G_{l,m}^{(n)}$ associated to the weight $\omega\in \mathcal{P}_{n-1}^{1}$ is a polynomial ensemble with
			\begin{equation}\label{Polya.G}
			w_b(a_c)=(-a_c\partial_c)^{b-1}\omega(a_c),
			\end{equation}
			i.e.,
			\begin{equation}\label{Polya.G.b}
			p_A(a)=\frac{1}{\prod_{j=1}^{n} j!\mathcal{M}\omega(j)}\Delta_n(a)\det[(-a_c\partial_c)^{b-1}\omega(a_c)]_{b,c=1,\ldots,n}.
			\end{equation}
\item		A polynomial ensemble on $H_m^{(n)}$ associated to the weights $w_1,\ldots,w_n\in L_{n-1}^1(\mathbb{R})$ is a $K$-invariant ensemble whose eigenvalues $a\in D$ are distributed as
			\begin{equation}\label{pol.H}
			p_D(a)=\frac{1}{n!}\Delta_n(a)\frac{\det[w_b(a_c)]_{b,c=1,\ldots,n}}{\det[\mathcal{M}w_b(c,c-1)]_{b,c=1,\ldots,n}}\in L_{\rm Prob}^{1}(D_n).
			\end{equation}
\end{enumerate}
\end{definition}

The true potential of the ensembles above unfolds itself when looking at their spherical transform that take particularly simple forms.

\begin{proposition}[Spherical Transforms of Polynomial and P\'olya Ensembles]\label{prop:spher.ensembles}\

\begin{enumerate}
\item		Let $P_G$ the distribution of a polynomial ensemble on $G_{l,m}^{(n)}$ associated to the weights $w_1,\ldots,w_n\in L_{n-1}^1(\mathbb{R}_+)$. Then, its spherical transform is (see~\cite{KK2016,KK2019} for the square case)
			\begin{equation}\label{spher.pol.G}
			\mathcal{S}_\Psi P_G(s)=\left(\prod_{j=0}^{n-1}j!\right)\frac{\det[\mathcal{M}w_b(s_c+1)]_{b,c=1,\ldots,n}}{\Delta_n(s)\det[\mathcal{M}w_b(c)]_{b,c=1,\ldots,n}},
			\end{equation}
			where $\mathcal{M}$ is the univariate Mellin transform on $\mathbb{R}_+$.
\item		The spherical transform of the distribution $P_G$, which is a P\'olya ensemble on $G_{l,m}^{(n)}$ associated to the weight $\omega\in \mathcal{P}_{n-1}^{1}$, is explicitly given by (see~\cite{KK2016,KK2019} for $l=m=n$)
			\begin{equation}\label{spher.Polya.G}
			\mathcal{S}_\Psi P_G(s)=\prod_{j=1}^{n}\frac{\mathcal{M}\omega(s_j+1)}{\mathcal{M}\omega(n-j+1)}.
			\end{equation}
\item		The spherical transform of  $P_H$ describing a polynomial ensemble on $H_m^{(n)}$ associated to the weights $w_1,\ldots,w_n\in L_{n-1}^1(\mathbb{R})$ is equal to
			\begin{equation}\label{spher.pol.H}
			\mathcal{S}_\Phi P_H(s,L)=\left(\prod_{j=0}^{n-1}j!\right)\frac{\det[\mathcal{M}w_b(s_c+1,L_c)]_{b,c=1,\ldots,n}}{\Delta_n(s)\det[\mathcal{M}\omega(c,c-1)]_{b,c=1,\ldots,n}}
			\end{equation}
			with $\mathcal{M}$ being the univariate Mellin transform on $\mathbb{R}$, see Eq.~\eqref{Mellin}.
\end{enumerate}
\end{proposition}

\begin{proof}
The first two parts are proven in~\cite{KK2016,KK2019} and the third claim is also straightforward. We only plug the definition~\eqref{pol.H} into Eq.~\eqref{Strafo:as}, apply Andr\'eief's identity~\cite{An86} and identify the integral in the determinant with the Mellin transform~\eqref{Mellin} on $\mathbb{R}$.
\end{proof}

An interesting consequence of the second part of Proposition~\ref{prop:spher.ensembles} is a relation between the spherical transforms of the P\'olya ensembles on $G_{l,m}^{(n)}$   associated to the weight $\omega\in \mathcal{P}_{n-1}^{1}$ and on $G_{l',m'}^{(n')}$ with the weight $\omega'(a)=a^{n-n'}\omega(a)$, where we assume $n>n'$. This relation reads
\begin{equation}
S_\Psi P_{G,\omega}({\rm diag\,}(s+(n-n')\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_{n-n'},s^{(n-n')})=S_\Psi P_{G,\omega'}(s).
\end{equation}
We will apply this observation several times in the ensuing sections.

It is a legitimate question to ask whether there is something like a P\'olya ensemble on $H_m^{(n)}$. Indeed, one can choose those ensembles induced by $x=\pm gg^*$ with $g\in G_{m,n}^{(n)}$ where $g$ is drawn from a P\'olya ensemble on $G_{m,n}^{(n)}$. However, one gets the feeling that these are the only realizations when looking for other ensembles. We have seen via trial and error that the positivity condition does not work well with the integrability when the support of the eigenvalues lies on both sides of the real axis. Since a proof is lacking let us phrase the above observation as a conjecture.

\begin{conjecture}[P\'olya ensembles on $H_m^{(n)}$]\label{conj:Polya}\

The only polynomial ensembles on $H_m^{(n)}$ that have the form $w_b(s_c)=(-s_c\partial_c)^{b-1}\omega(s_c)$ in Eq.~\eqref{pol.H} are those induced by $x=\pm gg^*$ with $g$ a random matrix drawn from a P\'olya ensemble on $G_{m,n}^{(n)}$.
\end{conjecture}

The reason why we care about P\'olya ensembles is the structural extremely simple form of their spherical transform, cf., Eq.~\eqref{spher.Polya.G}. It allows immediate conclusions for the convolutions~\eqref{conv.G} and~\eqref{conv.H}. For instance convolutions on $G_n$ of P\'olya ensembles are closed and have a semi-group action on polynomial ensembles on $G_n$, see~\cite{KK2019}. We will see that the P\'olya ensemble on $G_{l,m}^{(n_1)}$ have also a natural action on polynomial ensembles on $H_m^{(n_2)}$.

\begin{example}[Projections/Inclusions Revisited]\label{ex:proj}\

To generalize this to $G_{l,m}^{(n)}$ and $H_m^{(n)}$, we need to combine Proposition~\ref{prop:spher.ensembles} with the factorization Proposition~\ref{cor:fact:trans}. There the identification of the weight that creates the constant $C_{M,m}({\rm diag\,}(s+(m-l)\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_l,s^{(m-l)}))$ with $l\leq m\leq M$, see Eq.~\eqref{spher.uni}, is crucial. The good thing is that it corresponds to the weight
			\begin{equation}\label{omegal}
			\omega_{M-m}^{(m-l)}(a)=\left\{\begin{array}{cl} \displaystyle (M-m)a^{m-l}(1-a)^{M-m-1}\Theta(1-a), & M>m,\\ \delta(a-1), & M=m. \end{array}\right.
			\end{equation}
			The prefactor $(M-m)$ in the first case guarantees that the limit $M\to m$ yields the Dirac delta function in the second case  in the sense of weak topology. If $M\geq m+n$, this weight even satisfies the differentiability condition so that it can be associated to a P\'olya ensemble. This condition is exactly the one needed in~\cite{KKS2016} when considering products involving truncated unitary matrices.
\end{example}

Henceforth, we are more interested in the convolution~\eqref{conv.H} instead of Eq.~\eqref{conv.G}. The latter has been discussed extensively in several works over the past years~\cite{Kuijlaars2016,CKW2015,AI2015,KK2019,FKK2017}. Especially, we aim at computing the change of the bi-orthonormal functions $\{p_j,q_j\}_{j=0,\ldots,n-1}$ of the corresponding polynomial ensembles. Let us recall what these bi-orthonnormal functions are for a polynomial ensemble associated to the weights $\{w_j\}_{j=0,\ldots,n-1}$. The functions $\{p_j,q_j\}_{j=0,\ldots,n-1}$ satisfy three conditions, namely that $\{p_j\}_{j=0,\ldots,n-1}$ and $\{q_j\}_{j=0,\ldots,n-1}$  are bases of the linear span of the monomials $\{x^j\}_{j=0,\ldots,n-1}$ and of the weights $\{w_j\}_{j=1,\ldots,n}$, respectively, and additionally respect the bi-orthonormality condition
\begin{equation}\label{biortho}
\int da p_j(a)q_i(a)=\delta_{j,i}\ \text{for all}\ i,j=0,\ldots,n-1.
\end{equation}
Once these bi-orthonormal functions are given, it is straightforward to construct the kernel
\begin{equation}\label{kernel}
K_n(a_1,a_2)=\sum_{j=0}^{n-1}p_j(a_1)q_j(a_2)
\end{equation}
that determines all $\kappa$-point correlation functions
\begin{equation}\label{kpoint}
R_\kappa(a_1,\ldots a_{\kappa})=\frac{n!}{(n-\kappa)!}\prod_{j=\kappa+1}^n\int da_j p(a)=\det[K_n(a_b,a_c)]_{b,c=1,\ldots,\kappa}.
\end{equation}
The latter comprises all spectral statistical information of the random matrix. We would like to emphasize that the bi-orthonormal functions are not unique. The various choices can be exploited by picking the one suited the best for the considered problem. Additionally, one can specify a polynomial ensemble, either on $G_{l,m}^{(n)}$ or on $H_m^{(n)}$, by its bi-orthonormal functions $\{p_j,q_j\}_{j=0,\ldots,n-1}$ instead of the associated weights $\{w_j\}_{j=1,\ldots,n-1}$. We will exploit this when multiplying P\'olya ensembles on $G_{l,m}^{(n_1)}$ with polynomial ensembles on $H_m^{(n_2)}$ in Subsection~\ref{sec:mult.H}. Therein, the goal will be to understand the change of the bi-orthonormal functions.

\subsection{Multiplication of a Fixed Matrix on $H_m^{(n)}$}\label{sec:mult.G}

First, we would like to consider the case where $x\in H_m^{(n_2)}$ is fixed and $g\in G_{l,m}^{(n_1)}$ is a P\'olya ensemble.  Such a random matrix has the following joint probability densities of its eigenvalues.

\begin{theorem}[JPDF of a P\'olya Ensemble on $G_{l,m}^{(n_1)}$ Multiplied to a Fixed Matrix in $H_m^{(n_2)}$]\label{thm:jpdf.fixed}\

We choose four integers $l,m, n_1,n_2$ satisfying the two conditions $l,m\leq n_1$ and $m\leq n_2$. Moreover, we draw a random matrix $g\in G_{l,m}^{(n_1)}$ described by a P\'olya ensemble associated to the weight $\omega\in \mathcal{P}_{n-1}^{1}$ and $x\in H_m^{(n_2)}$ fixed with $a\in D_n$ its non-zero and non-degenerate eigenvalues. Then, we have the two cases:
\begin{enumerate}
\item		$n_1\geq n_2$:	Then the eigenvalues $\tilde{a}$ of the product $gag^*$ are distributed by
\begin{equation}\label{pfixed.g}
p(\tilde{a}|a)=\frac{1}{n_2!}\left(\prod_{j=1}^{n_2}\frac{(m-j)!}{(m-n_1)!(n_1-j)!\mathcal{M}\omega(n_1-j+1)}\right)\frac{\Delta_{n_2}(\tilde{a})}{\Delta_{n_2}(a)}\det\left[\tilde{\omega}_\geq(\tilde{a}_b|a_c)\right]_{b,c=1,\ldots,n_2}
\end{equation}
with the weight
\begin{equation}\label{weight.fixed.g}
\tilde{\omega}_\geq(\tilde{a}_b|a_c)=\frac{1}{|a_c|}\tilde{\omega}_\geq\left(\left.\frac{\tilde{a}_b}{a_c}\right|1\right)=\Theta(\tilde{a}_b a_c)\left(\frac{\tilde{a}_b}{a_c}\right)^{n_1-n_2}\int_0^\infty \frac{d a'}{|a_c|a'}\omega_{m-n_1}^{(0)}(a')\omega\left(\frac{\tilde{a}_b}{a_ca'}\right),
\end{equation}
where $\omega_{m-n_1}^{(0)}$ is defined in Eq.~\eqref{omegal} and $\Theta$ the Heaviside step function. The integral is evaluated at $a'=1$ when $m=n_1$.
\item		$n_1< n_2$: In this case the joint probability distribution of the eigenvalues $\tilde{a}$ of the product $gag^*$ is
\begin{equation}\label{pfixed.l}
\begin{split}
p(\tilde{a}|a)=&\frac{1}{n_1!}\left(\prod_{j=1}^{n_1}\frac{(m-j)!}{(m-n_1)!(n_1-j)!\mathcal{M}\omega(n_1-j+1)}\right)\frac{\Delta_{n_1}(\tilde{a})}{\Delta_{n_2}(a)}\det\left[\begin{array}{c} a_c^{b-1} \\ \tilde{\omega}_<(\tilde{a}_d|a_c) \end{array} \right]_{\substack{b=1,\ldots, n_2-n_1 \\ d=1,\ldots,n_1 \\ c=1,\ldots, n_2}}\\
=&\frac{1}{n_1!}\left(\prod_{j=1}^{n_1}\frac{(m-j)!}{(m-n_1)!(n_1-j)!\mathcal{M}\omega(n_1-j+1)}\right)\Delta_{n_1}(\tilde{a})\det\left[\sum_{c=1}^{n_2} \frac{e_{n_1-b}(-a_{\neq c})}{\prod_{h\neq c}(a_c-a_h)}\tilde{\omega}_<(\tilde{a}_d|a_c)\right]_{b,d=1,\ldots,n_1},
\end{split}
\end{equation}
with $a_{\neq j}={\rm diag\,}(a_1,\ldots,a_{j-1},a_{j+1},\ldots,a_{n_2})\in D_{n_2-1}$, the weight
\begin{equation}\label{weight.fixed.l}
\tilde{\omega}_<(\tilde{a}_d|a_c)=\frac{a_c^{n_2-n_1}}{|a_c|}\tilde{\omega}_<\left(\left.\frac{\tilde{a}_d}{a_c}\right|1\right)=\Theta(\tilde{a}_d a_c)a_c^{n_2-n_1}\int_0^\infty \frac{d a'}{|a_c|a'}\omega_{m-n_1}^{(0)}(a')\omega\left(\frac{\tilde{a}_d}{a_ca'}\right)
\end{equation}
and the elementary symmetric polynomials of order $o\leq n_2-1$ of $n_2-1$ elements
\begin{equation}\label{elementary}
e_o(-a_{\neq j})=\oint\frac{dz}{2\pi i z^{n_2-o}} \prod_{h\neq j}(z-a_h),
\end{equation}
where the contour is taken counter-clockwise around the origin.
\end{enumerate}
In particular, in both cases the random matrix $g\tilde{a} g^*$ is equal to a polynomial ensemble on $H_l^{(\min\{n_1,n_2\})}$. For degenerate spectra of $x$ one needs to perform l'H\^opital's rule.
\end{theorem}

The joint probability density is the starting point in deriving the spectral statistics in terms of its kernel. For this purpose it is crucial to find the corresponding bi-orthonormal functions. The good thing is that we have already well-prepared the weight for that in Theorem~\ref{thm:jpdf.fixed}.

\begin{corollary}[Eigenvalue Statistics of Products of P\'olya Ensembles with Fixed Matrices]\label{cor:stat.fixed}\

We consider the same setting as in Theorem~\ref{thm:jpdf.fixed}. Additionally, we define
\begin{equation}\label{orth.fixed}
\begin{split}
				p_j(\tilde{a})=&a^j,\\
				q_j^{\geq}(\tilde{a})=&\frac{(m+j-n_2)!}{(m-n_1)!(n_1+j-n_2)!}\sum_{c=1}^{n_2} \frac{e_{n_2-j-1}(-a_{\neq c})}{\prod_{h\neq c}(a_c-a_h)}\frac{\tilde{\omega}_\geq(\tilde{a}|a_c)}{\mathcal{M}\omega(n_1+j-n_2+1)},\\
				q_j^{<}(\tilde{a})=&\frac{(m+j-n_1)!}{(m-n_1)!j!}\sum_{c=1}^{n_2} \frac{e_{n_1-j-1}(-a_{\neq c})}{\prod_{h\neq c}(a_c-a_h)}\frac{\tilde{\omega}_<(\tilde{a}|a_c)}{\mathcal{M}\omega(j+1)}.
\end{split}
\end{equation}
Then,  the functions $\{p_j,q_j^\geq \}_{j=0,\ldots,n_2-1}$  are the bi-orthonormal set for $n_1\geq n_2$ and $\{p_j,q_j^<\}_{j=0,\ldots,n_1-1}$ is it for $n_1< n_2$. Hence, the kernels are given as
\begin{equation}\label{kern.fixed.g}
K_{n_2}^{\geq}(\tilde{a}_1,\tilde{a}_2)=\sum_{j=0}^{n_2-1}\sum_{c=1}^{n_2}\frac{(m+j-n_2)!}{(m-n_1)!(n_1+j-n_2)!}\frac{e_{n_2-j-1}(-a_{\neq c})}{\prod_{h\neq c}(a_c-a_h)}\frac{\tilde{a}_1^{j-1}\ \tilde{\omega}_\geq(\tilde{a}_2|a_c)}{\mathcal{M}\omega(n_1+j-n_2+1)}
\end{equation}
for $n_1\geq n_2$ and
\begin{equation}\label{kern.fixed.l}
K_{n_1}^<(\tilde{a}_1,\tilde{a}_2)=\sum_{j=0}^{n_1-1}\sum_{c=1}^{n_2}\frac{(m+j-n_1)!}{(m-n_1)!j!}\frac{e_{n_1-j-1}(-a_{\neq c})}{\prod_{h\neq c}(a_c-a_h)}\frac{\tilde{a}_1^{j-1}\ \tilde{\omega}_<(\tilde{a}_2|a_c)}{\mathcal{M}\omega(j+1)}
\end{equation}
for $n_1< n_2$.
\end{corollary}

\begin{proof}
For $n_1\geq n_2$ the statement above immediately follows from the computation
\begin{equation}\label{proof.fixed.1}
\int_{-\infty}^\infty d\tilde{a}\ \tilde{a}^{b}q_{b'}^\geq(\tilde{a})=\sum_{c=1}^{n_2} \frac{e_{n_2-b'-1}(-a_{\neq c})}{\prod_{h\neq c}(a_c-a_h)} a_c^{b}=\delta_{bb'},
\end{equation}
where the last equality has been shown in Eq.~\eqref{elem.orth}. Moreover, the functions $q_j$ are obviously a linear combination of the original weights $\{\tilde{\omega}_\geq(\tilde{a}|a_c)\}_{c=1,\ldots,n_2}$, which is everything to be shown. In exactly the same way one shows the claim for $n_1<n_2$.
\end{proof}

It is interesting that the structural form of the statistics for the embedding of a matrix from $x$ to $gxg^*$ with $n_1>n_2$ is not that much different from a projection ($n_1<n_2$). This would hint to an intimate relation between both operations albeit the statistics are certainly different. While for $n_1>n_2$ the number of non-zero eigenvalues stays the same, the rank of the matrix  decreases for $n_1<n_2$.

Another point we would like to highlight is  the representation of the kernels~\eqref{kern.fixed.g} and~\eqref{kern.fixed.l} as sums that can also be written in contour integrals under similar conditions as done in~\cite{Kieburg2017} for additive convolutions on matrix spaces. We will omit them here and go over to choosing the matrix $x\in H_{m}^{(n_2)}$ also randomly. Indeed for the case of $g$ being a product of Ginibre matrices this has been achieved in~\cite{Liu2017}.

\subsection{Multiplication of a Polynomial Ensemble on $H_m^{(n)}$}\label{sec:mult.H}

As before the joint probability density will be the starting point of our analysis.

\begin{theorem}[JPDF of a P\'olya Ensemble on $G_{l,m}^{(n_1)}$ Multiplied to a Polynomial Ensemble on in $H_m^{(n_2)}$]\label{thm:jpdf.poly}\

We consider the situation in Theorem~\ref{thm:jpdf.fixed} except that $x\in H_m^{(n_2)}$ is drawn from a polynomial ensemble associated to the weights $w_1,\ldots,w_n\in L_{n-1}^1(\mathbb{R})$. Again, we have to do the following case discussion:
\begin{enumerate}
\item		$n_1\geq n_2$:	The eigenvalues $\tilde{a}$ of the random  matrix product $gxg^*$ are distributed by
\begin{equation}\label{ppoly.g}
p(\tilde{a})=\frac{1}{n_2!}\left(\prod_{j=1}^{n_2}\frac{(m-j)!}{(m-n_1)!(n_1-j)!\mathcal{M}\omega(n_1-j+1)}\right)\Delta_{n_2}(\tilde{a})\frac{\det\left[\int_{-\infty}^\infty da \tilde{\omega}_\geq(\tilde{a}_b|a)w_c(a)\right]_{b,c=1,\ldots,n_2}}{\det\left[\mathcal{M}w_b(c,c-1)\right]_{b,c=1,\ldots,n_2}}
\end{equation}
where $\tilde{\omega}_\geq$ is given in Eq.~\eqref{weight.fixed.g}.
\item		$n_1< n_2$: The joint probability distribution of the eigenvalues $\tilde{a}$ of $gag^*$ is
\begin{equation}\label{ppoly.l}
\begin{split}
p(\tilde{a}|a)=\frac{1}{n_1!}\left(\prod_{j=1}^{n_1}\frac{(m-j)!}{(m-n_1)!(n_1-j)!\mathcal{M}\omega(n_1-j+1)}\right)\Delta_{n_1}(\tilde{a})\frac{\det\left[\begin{array}{c} \mathcal{M}w_c(b,b-1) \\ \int_{-\infty}^\infty da\tilde{\omega}_<(\tilde{a}_d|a)w_c(a) \end{array} \right]_{\substack{b=1,\ldots, n_2-n_1 \\ d=1,\ldots,n_1 \\ c=1,\ldots, n_2}}}{\det\left[\mathcal{M}w_b(c,c-1)\right]_{b,c=1,\ldots,n_2}},
\end{split}
\end{equation}
with the weight $\tilde{\omega}_<$ from Eq.~\eqref{weight.fixed.l}.
\end{enumerate}
\end{theorem}

\begin{proof}
The proof is straightforward since we only need to multiply Eqs.~\eqref{pfixed.g} and~\eqref{pfixed.l} with the probability density~\eqref{Polya.G.b} and integrate over $a\in D_{n_2}$. The Vandermonde determinant $\Delta_{n_2}(a)$ cancels and the remaining integral is carried out with the aid of Andr\'eief's identity~\cite{An86} leading to the statements.
\end{proof}

The second result~\eqref{ppoly.l} for $n_1<n_2$ as well as the first one~\eqref{ppoly.g} can be simplified drastically when choosing a set of bi-orthonormal functions $\{p_j,q_j\}_{j=0,\ldots,n_2-1}$ for the polynomial ensemble of $x\in H_m^{(n_2)}$. Firstly, the determinant $\det\left[\mathcal{M}w_b(c,c-1)\right]_{b,c=1,\ldots,n_2}$ in the  denominators is replaced by $\det\left[\int_{-\infty}^\infty da\,p_{b-1}(a)q_{c-1}(a)\right]_{b,c=1,\ldots,n_2}=1$ which equals unity. Secondly, the determinant in the numerator of Eq.~\eqref{ppoly.l} is essentially of size $n_1\times n_1$ since the first $n_2-n_1$ rows become Kronecker deltas. Summarizing, in the terms of the bi-orthonormal functions of the random matrix $x$ , the two equations~\eqref{ppoly.g} and~\eqref{ppoly.l} are equal to
\begin{equation}\label{ppoly.g.b}
p(\tilde{a})=\frac{1}{n_2!}\left(\prod_{j=1}^{n_2}\frac{(m-j)!}{(m-n_1)!(n_1-j)!\mathcal{M}\omega(n_1-j+1)}\right)\Delta_{n_2}(\tilde{a})\det\left[\int_{-\infty}^\infty da \tilde{\omega}_\geq(\tilde{a}_b|a)q_{c-1}(a)\right]_{b,c=1,\ldots,n_2}
\end{equation}
and
\begin{equation}\label{ppoly.l.b}
p(\tilde{a}|a)=\frac{1}{n_1!}\left(\prod_{j=1}^{n_1}\frac{(m-j)!}{(m-n_1)!(n_1-j)!\mathcal{M}\omega(n_1-j+1)}\right)\Delta_{n_1}(\tilde{a})\det\left[\int_{-\infty}^\infty da\tilde{\omega}_<(\tilde{a}_d|a)q_{n_2-n_1+c-1}(a) \right]_{c,d=1,\ldots,n_1},
\end{equation}
respectively.

 This simplification comes in handy when  analysing the transformations of the statistics, especially the set of bi-orthonormal functions, of the Hermitian random matrix $x\in H_m^{(n_2)}$ to the one  of $gxg^*\in H_l^{(r)}$ with $r=\min\{n_1,n_2\}$. There are two reason why this might be of interest. The first one is a practical one. As already explained the bi-orthonormal functions build up the whole spectral statistics for this class of ensembles. The second reason aims at a better understanding of the random matrix multiplication as a statistical process. We have formulated the ``analytical response" of such a multiplication in the following proposition.

\begin{proposition}[Eigenvalue Statistics of Products of P\'olya Ensembles with Polynomial Ensembles]\label{prop:stat.poly}\

We consider the setting of Theorem~\ref{thm:jpdf.poly} and assume that the polynomial ensemble of $x$ corresponds to the bi-orthonormal functions $\{p_j,q_j\}_{j=0,\ldots,n_2-1}$ with the kernel $K_{n_2}(a_1,a_2)$.
Moreover, we define the polynomials
\begin{equation}\label{chi.def}
\begin{split}
\chi_\geq(z)=&\sum_{j=0}^{n_2-1}\frac{(m+j-n_2)!}{(m-n_1)!(n_1+j-n_2)!\mathcal{M}\omega(n_1+j-n_2+1)}z^j,\\
\chi_<(z)=&\sum_{j=0}^{n_1-1}\frac{(m+j-n_1)!}{(m-n_1)!j!\mathcal{M}\omega(j+1)}z^{j+n_2-n_1}.
\end{split}
\end{equation}
 Then, the bi-orthogonal functions $\{\tilde{p}_j,\tilde{q}_j\}_{j=0,\ldots,r-1}$ and kernel $\tilde{K}_r(\tilde{a}_1,\tilde{a}_2)$ of the product $gxg^*$ with rank $r=\min\{n_1,n_2\}$ are
\begin{enumerate}
\item		$n_1\geq n_2$:
			\begin{equation}\label{stat.pol.g}
			\begin{split}
			\tilde{p}_j(\tilde{a})=&\oint \frac{dz}{2\pi i z}\chi_\geq(z)p_j\left(\frac{\tilde{a}}{z}\right),\\
			\tilde{q}_j(\tilde{a})=&\int_{-\infty}^\infty \frac{ da}{|a|}\tilde{\omega}_\geq\left(\left.\frac{\tilde{a}}{a}\right|1\right)q_j(a),\\
			\tilde{K}_{n_2}(\tilde{a}_1,\tilde{a}_2)=&\oint \frac{dz}{2\pi i z}\chi_\geq(z)\int_{0}^\infty  \frac{ da}{a}\tilde{\omega}_\geq\left(\left.a\right|1\right)K_{n_2}\left(\frac{\tilde{a}_1}{z},\frac{\tilde{a}_2}{a}\right),\\
			\end{split}
			\end{equation}
\item		$n_1<n_2$:
			\begin{equation}\label{stat.pol.l}
			\begin{split}
			\tilde{p}_j(\tilde{a})=&\tilde{a}^{n_1-n_2} \oint \frac{dz}{2\pi i z}\chi_<(z)p_{n_2-n_1+j}\left(\frac{\tilde{a}}{z}\right),\\
			\tilde{q}_j(\tilde{a})=&\int_{-\infty}^\infty  \frac{da}{|a|}a^{n_2-n_1}\tilde{\omega}_<\left(\left.\frac{\tilde{a}}{a}\right|1\right),\\
			\tilde{K}_{n_1}(\tilde{a}_1,\tilde{a}_2)=&\left(\frac{\tilde{a}_2}{\tilde{a}_1}\right)^{n_2-n_1}\oint \frac{dz}{2\pi i z}\chi_<(z)\int_{0}^\infty \frac{da}{a^{n_2-n_1+1}}\tilde{\omega}_<\left(\left.a\right|1\right)K_{n_2}\left(\frac{\tilde{a}_1}{z},\frac{\tilde{a}_2}{a}\right).\\
			\end{split}
			\end{equation}
			The contour integrals run counter-clockwise around the origin.
\end{enumerate}
\end{proposition}

The upper and lower limits for $\chi_\geq$ and $\chi_<$ can be chosen arbitrarily if more moments exist. Indeed it can be even taken to infinity if the series exists in a ring around the origin. This is the case, for instance, for the Ginibre ensemble where $\mathcal{M}\omega(s)=\Gamma(s)$ and for the Jacobi ensemble with  $\mathcal{M}\omega(s)=\Gamma(s+\nu)\Gamma(\mu+n)/\Gamma(s+\nu+\mu+n)$ for $\nu>-1$ and $\mu> 0$, see~\cite{KKS2016,KK2019}. The case of the product of an induced Ginibre ensemble whose weight is equal to $\omega(a)=x^\nu e^{-a}$ has been studied in~\cite{FIL2018}; in particular Theorem~\ref{thm:jpdf.poly} and Proposition~\ref{prop:stat.poly}, up to normalization, become Lemma 2 and Proposition 7 in~\cite{FIL2018}, respectively. The statements above generalize this discussion and avoids the non-compact group integrals encountered in~\cite{FIL2018} that are unknown for more general weights, e.g., for the Jacobi ensemble with $\omega(a)=a^\nu(1-a)^{\mu+n-1}\Theta(1-a)$, Cauchy-Lorentz ensemble with $\omega(a)=a^\nu/(1+a)^{\mu+n}$ or the Muttalib--Borodin ensembles like $\omega(a)=a^\nu e^{-a^\theta}$ with $\theta>0$ or  $\omega(a)=a^\nu e^{-({\rm ln}\,a)^2}$. All of these examples are P\'olya ensembles, and our theoretical framework and its results deals with them in a unifying way.

\begin{example}[Projections and Inclusions of Hermitian Matrices]\label{ex:appl}\

We would like to conclude this section with an example  highlighting that $\omega$ can be even a distribution. This example is given by a projection or an inclusion of the Hermitian matrix $x\in H_{m}^{(n_2)}$.

For the orthogonal projection we choose $g\in G_{l,m}^{(l)}$ (in particular $l\leq m$) to be an $l\times m$ block of a Haar distributed unitary matrix of size $M\times M$, see Example~\ref{ex:uni}. Then, we know from Eq.~\eqref{spher.uni} comparing with Eq.~\eqref{spher.Polya.G} that the corresponding Mellin transform of $\omega$ is given by
\begin{equation}
\mathcal{M}\omega(s)=\frac{\Gamma[M-m+1]\Gamma[s+m-l]}{\Gamma[s+M-l]}.
\end{equation}
This corresponds to the identification $\omega=\omega_{M-m}^{(m-l)}$, see Example~\ref{ex:proj}, which does not always satisfy the differentiability criterion for arbitrary $m,l\leq M$ leading to a distribution for the joint probability density. Nonetheless, we can apply the factorization of the spherical transform, see Corollary~\ref{cor:fact:trans}. This leads to the two cases
\begin{equation}
\begin{split}
\tilde{\omega}_\geq(\tilde{a}|a)=&\frac{(m-l)!(M-m)!}{(M-l)!}\frac{\Theta(\tilde{a} a)}{|a|}\omega_{M-l}^{(l-n_2)}\left(\frac{\tilde{a}}{a}\right),\quad \chi_\geq(z)=\sum_{j=0}^{n_2-1}\frac{(M+j-n_2)!}{(m-l)!(l+j-n_2)!(M-m)!}z^j
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
\tilde{\omega}_<(\tilde{a}|a)=&\frac{(m-l)!(M-m)!}{(M-l)!}\frac{\Theta(\tilde{a} a)}{|a|}\omega_{M-l}^{(0)}\left(\frac{\tilde{a}}{a}\right),\quad \chi_<(z)=\sum_{j=0}^{l-1}\frac{(M-l+j)!}{(m-l)!j!(M-m)!}z^{j+n_2-l}
\end{split}
\end{equation}
depending on whether $l\geq n_2$ or $l< n_2$, respectively. An orthogonal projection in the original sense is given by $m=M$ since the unitary matrix can be absorbed into $x$ due to the $K$-invariance of the latter. Thus it reduces to a projection onto the first $l$ rows. For $l=m=M$ we notice that the functions naturally reduce to those that keep the statistics the same.

For the inclusion we consider $l\geq m$ with $g\in G_{l,m}^{(m)}$. This leads to the Mellin transform of $\omega$
\begin{equation}
\mathcal{M}\omega(s)=\frac{\Gamma[M-l+1]\Gamma[s+l-m]}{\Gamma[s+M-m]}.
\end{equation}
which corresponds to $\omega=\omega_{M-l}^{(l-m)}$.  Since there is only one situation to consider namely $l\geq m\geq n_2$, the constant in front of Corollary~\ref{cor:fact:trans} is equal to unity ($C_{m,m}(\tilde{s})=1$). Therefore the two functions in the transformation of the eigenvalue statistics are
\begin{equation}
\begin{split}
\tilde{\omega}_\geq(\tilde{a}|a)=&\frac{\Theta(\tilde{a} a)}{|a|}\omega_{M-l}^{(l-m)}\left(\frac{\tilde{a}}{a}\right),\quad \chi_\geq(z)=\sum_{j=0}^{n_2-1}\frac{(M+j-m)!}{(l+j-m)!(M-l)!}z^j.
\end{split}
\end{equation}
A true random inclusion, that is unitarily invariant, would be the case $M=l$ which interestingly does not change the statistics of $x$ to $gxg^*$ at all. Indeed it becomes immediately clear why it is so when considering the characteristic polynomial where we have
\begin{equation}
\det(gxg^*-a\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_M)=\det(\Pi_{M,m}x\Pi_{m,M}-a\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_M)=\det(x-a\leavevmode\hbox{\small1\kern-3.8pt\normalsize1}_M)
\end{equation}
for any $a\in\mathbb{C}$. In the first equality, we have exploited the fact $g$ can be written as product of unitary matrix and the projection $\Pi_{M,m}$ when $M=l$.
\end{example}

\section{Conclusions}\label{sec:conclusio}

We extended the harmonic analysis approach to products of complex rectangular matrices and Hermitian matrices of a fixed rank. As already experienced for products of real asymmetric and real anti-symmetric matrices~\cite{KFI2019}, we need two different spherical functions and, hence, transforms, one for each matrix space. They slightly differ from the original definition by Harish-Chandra et al., see~\cite{Helgason2000} and references therein, in the normalization and the duplication of the complex plane of the ``Mellin-Fourier'' (frequency) parameter $s$. In total, one needs $2^r$ copies for a matrix of rank $r$ denoted by the vector $L\in\mathbb{Z}_2^r\simeq\{0,1\}^r$. These copies are essential to keep the information of the number of positive and negative eigenvalues, since this number stays fixed in such a product as already observed in~\cite{FIL2018}. In spite of these two modifications of the spherical transform, the results resemble very much the original results for matrices on the general (special) linear group~\cite{Helgason2000,KK2016}.

We applied this theoretical framework to a product of a P\'olya ensemble on the complex rectangular matrices and a
fixed Hermitian matrix as well as a random Hermitian matrix drawn from a polynomial ensemble. All matrices have a specific rank that does not need to be maximal. Both cases yield again polynomial ensembles.  We computed their joint probability densities of the eigenvalues, sets of bi-orthonormal functions and their kernel. For the case of a random Hermitian matrix it is noteworthy to say that the contour integral representations of the bi-orthonormal functions and kernels look extremely similar to those already found for products of complex matrices~\cite{KK2019} and sums on the classical Lie-algebras~\cite{Kieburg2017}. Those formulas are an ideal basis to start a large $n$-analysis. A work on the hard edge statistics of the products of multiplicative P\'olya ensembles with a GUE is currently in preparation~\cite{Kieburg2019}.

What is still puzzling, even disturbing, is the rather different generalization of the spherical function when comparing the case of real antisymmetric matrices~\cite[Equation~(2.11)]{KFI2019} and of Hermitian matrices, see Eq.~\eqref{spher:as}. In the former we only  omitted each second frequency $s_j$, since they correspond to vanishing determinants, while in the latter we even needed to extend the frequency space to an additional parameter set $L$. Thus, harmonic analysis on specific representations of Lie groups seems to avoid a simple unified approach. Regarding this point, we should mention that the corresponding harmonic analysis for the adjoint action of the general linear groups on  the Lie algebras of the orthogonal matrices of odd dimension and of unitary symplectic matrices are still open. Maybe when these gaps are filled, one can easier identify the proper framework that encompasses all these cases and, hopefully, even more.

\section*{Acknowledgements}

I am grateful for the fruitful discussions with Holger K\"osters. Additionally, I would like to thank Peter Forrester and Dang-Zheng Liu for reading the first draft of this work.

redpajama_set_name: RedPajamaArXiv


I first discovered yoga when I bought a couple of Kundalini yoga DVDs off Amazon while living overseas in 2006. Those two DVDs were my constant while living in Europe for four years, away from family and suffering from severe anxiety and depression.
When I came back to New Zealand in 2010 yoga fell by the wayside. I tried a couple of studios over the years but nothing motivated me to keep going. When I saw a new studio was opening in Wellington I thought I'd try a class on opening day. As soon as I walked into the Power Living studio I instantly felt a sense of community and aroha. I signed up for a year on the spot and have never looked back.
I love every aspect of the practice; the way I feel after a strong asana class, how the 5000 year old philosophy is still so relevant to our modern lives, and how our muscles, organs and nervous system work and respond to movement and meditation. I have a thirst for learning new things and integrating them into my life, and I can't wait to share with my community and increase the yoga love!
I love crow pose (kakasana). There is a balance between strength and faith that just makes me feel so strong and free.
To me, being a modern yogi is about exploring and understanding the traditional philosophy and finding ways to apply it to the crazy world we live in. Our modern society is completely different to 5000 years ago when yoga is said have originated, so for me, being an example of someone who is compassionate, lives in their power and continues to look inside to inspire change in others is what being a modern yogi means to me.


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text:
Fans gather from around the world to celebrate Elvis Presley's birthday at Graceland
By Kelly Roberts | January 8, 2020 at 12:45 PM CST - Updated January 8 at 6:28 PM
MEMPHIS, Tenn. (WMC) - Elvis Presley would have been 85 years old Wednesday. Year after year thousands of Presley fans flock to Memphis to say thank you, thank you very much to the King on his birthday.
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It's at Graceland where Priscilla Presley said Elvis spent most of his birthdays. They were much smaller affairs then, but no matter how small or large the celebrations are everyone is here to say thanks to Elvis.
If you would like to score some Elvis Presley goods, visit https://www.ebay.com/e/_collectibles/elvis-85th-birthday.
Kelly Roberts
Kelly Roberts grew up in the Hoosier State. She has reported on presidential visits, a deadly state fair stage collapse and two school shootings. In her free time, Kelly enjoys traveling, reading, watching Netflix and playing with her cat, Custard.
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