text:
\section{Introduction}
In this article we provide discrete characterizations of wave front sets of various types. We shall show that the microlocal properties of (ultra)distributions are completely determined  by the decay properties of the restrictions of Fourier transforms of their localizations to an arbitrary lattice in $\mathbb{R}^{d}$. To this end, we also study Fourier series expansions of ultradistributions. The paper refines and extends earlier results on \emph{toroidal} wave front sets from \cite{Maksi,RT}.

Wave front sets play a fundamental role in the analysis of propagation of singularities of solutions to partial differential equations. The classical wave front set, originally introduced by H\"{o}rmander \cite{Hormander1971}, is defined with respect to $C^{\infty}$-smoothness, but this concept can be refined to include wave front sets with respect to other smoothness scales, such as Denjoy-Carleman classes and, in particular, analyticity  \cite{Hormander, Komatsutwee, Rodino}. More recently, wave front sets with respect to Fourier-Lebesgue spaces and other classes of Banach and Fr\'{e}chet spaces have been introduced and studied in \cite{c-j-t1,c-j-t2,Pilipovicthree,Pilipovictwo}. All of these variants have been systematically applied to the study of regularity properties of various classes of pseudo-differential operators and semilinear equations. The notion of wave front set has also relevant applications in mathematical physics, see the expository article \cite{b-d-h} for an overview.

The question of whether the wave front set of a distribution can be described in a discrete fashion goes back to Ruzhansky and Turunen \cite{RT}. Naturally, this is a very important question from a computational point of view.
Their work is motivated by the study of (global) quantization of periodic pseudo-differential operators through Fourier series \cite{RT,Ruzh}. Denoting as $WF^{\mathbb{T}^{d}}(f)$ the ($C^{\infty}$) toroidal wave front set of a distribution $f$ on the $d$-dimensional torus $\mathbb{T}^{d}$, they have established the equality \cite[Sect. 7] {RT}
\begin{equation}
\label{RTwavefront}
WF^{\mathbb{T}^{d}}(f)=WF(f)\cap(\mathbb{T}^{d}\times \mathbb{Z}^{d}),
\end{equation}
where $WF(f)$ stands for the classical H\"{o}rmander wave front set when regarding $f$ as a distribution on $\mathbb{R}^{d}$. The latter equality further extends to Sobolev-type and Gevrey wave front sets, as recently shown in \cite{d-m-s,Maksi}. It should also be mentioned that Rodino and Wahlberg \cite{Rodino-W} and Johansson et al. \cite{Johansson} have investigated discrete definitions of microregularity properties of distributions via Gabor frames.

Our goal here is to generalize (\ref{RTwavefront}) in several directions. On the one hand, we prove that the equality (\ref{RTwavefront}) remains valid for wave front sets of Fourier-Lebesgue and quasianalytic types. On the other hand, we replace $\mathbb{Z}^{d}$ in (\ref{RTwavefront}) by an arbitrary lattice in $\mathbb{R}^{d}$. Since we will actually reformulate the equality (\ref{RTwavefront}) in slightly different terms, namely, in terms of discretized estimates for Fourier transforms, the arbitrariness of the lattice strengthens the potential computational content of the results.

We now briefly describe the content of the paper and state some samples of our results. We mention that we will work with both the Beurling-Bj\"{o}rck \cite{Bjorck} and the Komatsu \cite{Komatsu} approach to the theory of ultradifferentiable functions and ultradistributions (see the preliminary Section \ref{preliminaries} for the notation). It turns out that many of the arguments employed in the article depend upon the use of Fourier series of ultradistributions with respect to a lattice. In the case of $\omega$-ultradistributions, to the best of our knowledge, this topic is not available in the literature. Section \ref{periodic ultradistributions} gives a concise presentation of the theory of Fourier series expansions of periodic ultradistributions.

Section \ref{Section F-L} is dedicated to discrete characterizations of the Fourier-Lebesgue wave front set of a non-quasianalytic $\omega$-ultradistribution with respect to a so-called $\omega$-moderate weight. In particular, our considerations apply to the wave front sets $WF_{(\omega)}$ and $WF_{\{\omega\}}$. It is worth mentioning that the latter two classes of wave front sets have been recently studied by Albanese et al. \cite{a-j-o} and Fern\'{a}ndez et al. \cite{f-g-j} in connection with regularity of solutions to linear PDE. Note that wave front sets with respect to Fourier-Lebesgue spaces were originally introduced in  \cite{j-p-t-t,Pilipovicthree,Pilipovictwo}, but we remark that here we allow weights with much larger growth than those considered in the forementioned works. In addition, our results apply to more general classes of ultradistributions. The authors believe that the Beurling-Bj\"{o}rck theory is the most natural framework for microlocal analysis based on Fourier-Lebesgue spaces.

Our results from Section \ref{Section F-L} already include discrete characterizations of Gevrey $\{s\}$- and $(s)$-microregularity for $s>1$ \cite{Rodino}. Moreover, they also contain the case of microregularity with respect to certain classes of non-quasianalytic weight sequences. In Section \ref{section quasianalytic} we further extend our analysis to weight sequences satisfying milder assumptions. In the non-quasianalytic case, we shall show the following theorem. A lattice $\Lambda$ in $\mathbb{R}^{d}$ is simply a discrete subgroup of $\mathbb{R}^{d}$ which spans the real vector space $\mathbb{R}^{d}$. The dual lattice of $\Lambda$ is the discrete group $\Lambda^{\ast}=\{\mu^{\ast}\in\mathbb{R}^{d}:\: \mu\cdot \mu^{\ast}\in\mathbb{Z}, \:\forall \mu\in\Lambda\}$.

\begin{theorem}
\label{th WF non-quasianalytic} Let $(M_p)_{p\in\mathbb{N}}$ be a weight sequence satisfying the conditions $(M.1)$, $(M.2)'$, and $(M.3)'$ and having associated function $M$ (see Subsection \ref{subsection sequences}). Suppose that  $\Lambda$ is a lattice in $\mathbb{R}^{d}$ and let $U$ be an open convex neighborhood of the origin such that $U\cap\Lambda^{*}=\{0\}$. The Roumieu wave front set $WF_{\{M_p\}}(f)$ $($the Beurling wave front set $WF_{(M_p)}(f)$$)$ of an ultradistribution $f\in{\mathcal{D}^{(M_p)}}' (\Omega)$ coincides with the complement in $\Omega\times(\mathbb{R}^{d}\setminus\{0\})$ of the set of all points $(x_0,\xi_0)$ for which
there exist an open conic neighborhood $\Gamma$ of $\xi_0$ and $\varphi \in \mathcal{D}^{(M_p)}(\Omega\cap(x_0+U))$ with $\varphi\equiv1$ in a neighborhood of $x_0$  such that for some $r > 0$ $($for every $r>0$$)$
$$
\sup_{\mu \in \Gamma\cap \Lambda} |\widehat{\varphi f}(\mu)| e^{M(r\mu)} < \infty.
$$
\end{theorem}

Our main result from Section \ref{section quasianalytic}, Theorem \ref{mainqa}, actually covers quasianalytic wave front sets. Specializing Theorem \ref{mainqa} to the analytic wave front set of a distribution, one obtains:

\begin{theorem}
\label{th WF analytic} Let $\Lambda$ be a lattice in $\mathbb{R}^{d}$ and let $U$ be an open convex neighborhood of the origin such that $U\cap\Lambda^{*}=\{0\}$. The analytic wave front set $WF_{A}(f)$ of a distribution $f \in \mathcal{D}'(\Omega)$ coincides with the complement in $\Omega\times(\mathbb{R}^{d}\setminus\{0\})$ of the set of points $(x_0,\xi_0)$ for which there are an open neighborhood $V \subseteq \Omega\cap(x_0+U)$ of $x_0$,  an open conic neighborhood $\Gamma$ of $\xi_0$, and a bounded sequence $(f_p)_{p \in \mathbb{N}}$  in $\mathcal{E}'(\Omega\cap(x_0+U))$, with $f_p= f$ on $V$ for all $p \in \mathbb{N}$, such that for some $h > 0$
\[
  \sup_{\mu \in \Gamma \cap \Lambda} |\widehat{f_p} (\mu)| |\mu|^p  \leq h^{p+1}p!\:, \qquad \forall p \in \mathbb{N}.
   \]

\end{theorem}
\section{Preliminaries}
\label{preliminaries}
In this section we fix the notation and explain the spaces of ultradifferentiable functions and ultradistributions needed in this article.

\subsection{Spaces defined via weight functions}
\label{subsection weights} We start with the Beurling-Bj\"{o}rck approach to ultradistribution theory via weight functions \cite{Bjorck} (see also \cite{b-m-t}). A weight function on $\mathbb{R}^{d}$ is simply a non-negative measurable function. Throughout the article we shall always assume that $\omega$ is an even weight function satisfying $\omega(0)=0$ and the following three conditions:
\begin{itemize}
\item[$(\alpha)$]  $\omega(\xi_1 + \xi_2) \leq \omega(\xi_1) + \omega(\xi_2), \quad \forall \xi_1,\xi_2 \in \mathbb{R}^d,$
\item[$(\beta)$] $\displaystyle \int_{|\xi| > 1} \frac{\omega(\xi)}{|\xi|^{d + 1}} \mathrm{d}\xi < \infty,$
\item[$(\gamma)$] there exist $a \in \mathbb{R}$ and $C > 0$ such that $\omega(\xi) \geq a + C\log (1 + |\xi|), \quad \forall \xi \in \mathbb{R}^d.$
\end{itemize}
It should be noticed that measurability and subadditivity, namely, condition $(\alpha)$, ensure that $\omega$ is locally bounded \cite{beurling,hilleph}.

Let $\Omega \subseteq \mathbb{R}^d$ be open, $K \Subset\Omega$ (a compact subset in $\Omega$) and $\lambda>0$. The Banach space $\mathcal{D}_{\omega}^\lambda(K)$ consists of those $\varphi  \in L^{1}(\mathbb{R}^d)$ such that $\operatorname*{supp} \varphi \subseteq K$ and
\[\| \varphi \|_\lambda= \| \varphi \|_{\omega,\lambda}:= \sup_{\xi \in \mathbb{R}^d} |\widehat{\varphi}(\xi)|e^{\lambda \omega(\xi)} < \infty.\]
Set further
\[ \mathcal{D}_{(\omega)}(\Omega) = \varinjlim_{K \Subset \Omega} \varprojlim_{\lambda \to \infty} \mathcal{D}_{\omega}^\lambda(K).\]
Condition $(\gamma)$ yields  $\mathcal{D}_{(\omega)}(\Omega)\subseteq \mathcal{D}(\Omega)$. Its dual $\mathcal{D}_{(\omega)}'(\Omega)$ is the ultradistribution space of class $(\omega)$ (or  Beurling type). We define $\mathcal{E}_{(\omega)}(\Omega)$ as the space of multipliers of $\mathcal{D}_{(\omega)}(\Omega)$, that is, a function  $\varphi\in\mathcal{E}_{(\omega)}(\Omega)$ if and only if $\varphi\psi \in\mathcal{D}_{(\omega)}(\Omega)$ for all $\psi \in \mathcal{D}_{(\omega)}(\Omega)$. Its topology is generated by the family of seminorms
$ \varphi \to  \| \varphi\psi \|_{\lambda}$, $ \lambda > 0,$ $\psi \in \mathcal{D}_{(\omega)}(\Omega). $ Clearly,
$\mathcal{E}'_{(\omega)}(\Omega)$ is the subspace of $\mathcal{D}_{(\omega)}'(\Omega)$ consisting of ultradistributions with compact support. The space $\mathcal{S}_{(\omega)}(\mathbb{R}^d)$ consists of all those $\varphi\in C^\infty(\mathbb{R}^d)$ such that
\[\sup_{x \in \mathbb{R}^d}|\varphi^{(\alpha)}(x)|e^{\lambda \omega(x)} < \infty \quad\mbox{and}\quad \sup_{\xi \in \mathbb{R}^d}|\widehat{\varphi}^{(\alpha)}(\xi)|e^{\lambda \omega(\xi)} < \infty, \quad \forall\lambda > 0,\: \forall\alpha\in\mathbb{N}^{d};\]
its Fr\'{e}chet space topology being defined in the canonical way. We shall fix the constants in the Fourier transform as
\[ \mathcal{F} \varphi (\xi) = \widehat{\varphi}(\xi) = \int_{\mathbb{R}^d} \varphi(x) e^{-2\pi i \xi \cdot x} \mathrm{d}x. \]
Since the Fourier transform is an automorphism of  $\mathcal{S}_{(\omega)}(\mathbb{R}^d)$, it can be extended by duality to $\mathcal{S}_{(\omega)}'(\mathbb{R}^d)$, the so-called space of $(\omega)$-tempered ultradistributions.

Naturally, one may also define the spaces of class $\{\omega\}$  (Roumieu type)  $\mathcal{D}_{\{\omega\}}(\Omega)$, $\mathcal{D}'_{\{\omega\}}(\Omega)$, $\mathcal{E}_{\{\omega\}}(\Omega)$, $\mathcal{E}'_{\{\omega\}}(\Omega)$, $\mathcal{S}_{\{\omega\}}(\mathbb{R}^d)$, and $\mathcal{S}'_{\{\omega\}}(\mathbb{R}^d)$ by simply switching the universal quantifier $\forall$ over $\lambda$ to an existential one.
When considering these spaces, we shall always assume that $\omega$ satisfies a stronger condition than $(\gamma)$, namely,
\begin{itemize}\item[$(\gamma_{0})$] $\displaystyle\lim_{|\xi|\to\infty} \frac{\omega(\xi)}{\log (1 + |\xi|)}=\infty$.
\end{itemize}
For instance,
\[\mathcal{D}_{\{\omega\}}(\Omega) = \varinjlim_{K \Subset \Omega} \varinjlim_{\lambda \to 0^+} \mathcal{D}_{\omega}^\lambda(K),
\]
and $(\gamma_0)$ ensures that $\mathcal{D}_{\{\omega\}}(\Omega)\subsetneq \mathcal{D}(\Omega)$.

Note that if $\omega(\xi)=\log(1+|\xi|)$, one then recovers the classical Schwartz spaces as particular instances of the Beurling case. Another
important example of a weight function is provided by the Gevrey weights $\omega(\xi) = |\xi|^{\frac{1}{s}},$  $s>1$; in such a case one obtains the well known Gevrey function and ultradistribution spaces \cite{Rodino,j-p-t-t}.

We shall also work with weighted Fourier-Lebesgue spaces \cite[Chap. II]{Bjorck}. A weight function $v$ is said to be $(\omega)$-moderate  ($\{\omega\}$-moderate) if there are $C, \lambda > 0$ (for every $\lambda > 0$ there is $C=C_{\lambda}>0$) such that
\begin{equation}
v(\xi_1 + \xi_2) \leq Cv(\xi_1)e^{\lambda \omega(\xi_2)}, \quad \forall \xi_1,\xi_2 \in \mathbb{R}^d.
\label{moderate}
\end{equation}
The classes of all $(\omega)$-moderate and $\{\omega\}$-moderate weight functions are denoted by $\mathscr{M}_{(\omega)}$ and $\mathscr{M}_{\{\omega\}}$, respectively. Let $q \in [1, \infty]$; the (weighted) Fourier--Lebesgue space $\mathcal{F} L^q_v$ (with respect to $v \in \mathscr{M}_{(\omega)}$) is the Banach space of all $f \in \mathcal{S}_{(\omega)}'(\mathbb{R}^d)$ such that $\widehat{f} \in L^q_v(\mathbb{R}^{d})$, that is, $\widehat{f}$ is locally integrable and  $\| f\|_{\mathcal{F} L^q_{v}}:=\| \widehat{f}\|_{L^q_{v}}=\| \widehat{f}\: v\|_{L^q} < \infty$. Clearly, if $v\in\mathscr{M}_{\{\omega\}}$, then $\mathcal{F} L^q_v\subset  \mathcal{S}_{\{\omega\}}'(\mathbb{R}^d)$.

\subsection{Weight sequences}
\label{subsection sequences} Another useful and widely used approach to the theory of ultradifferentiable functions and ultradistributions is via weight sequences \cite{Komatsu}. Let $(M_p)_{p \in \mathbb{N}}$ be a sequence of positive real numbers (with $M_0=1$). We will make use of some of the following conditions:\begin{itemize}
\item [$(M.1)\:$] $M^{2}_{p}\leq M_{p-1}M_{p+1},$  $p\geq 1$,
\item [$(M.2)'$]$M_{p+1}\leq A H^p M_p$, $p\in\mathbb{N}$, for some $A,H>0$,
\item [$(M.2)\:$] $ \displaystyle M_{p}\leq A H^p\min_{1\leq q\leq p} \{M_{q} M_{p-q}\},$ $p\in\mathbb{N}$, for some $A,H>0$,
\item [$(M.3)'$] $\displaystyle \sum_{p=1}^{\infty}\frac{M_{p-1}}{M_{p}}<\infty. $
\end{itemize}
The associated function of $(M_p)_{p}$ is defined as
 $$M(t):=\sup_{p\in\mathbb{N}}\log\frac{t^p}{M_p},\quad t>0.$$
Its log-convex regularization is the sequence $\displaystyle M^{c}_{p}= \sup_{t>0}\frac{t^{p}}{e^{M(t)}}$, $p\in\mathbb{N}$, the greatest log-convex minorant of $(M_p)_{p}$; note that $(M_p)_{p}$ satisfies (M.1) if and only if $(M_p)_{p}=(M_p^{c})_{p}$ (see \cite{Komatsu}).  As usual, the relation $M_p\subset N_p$ between two weight sequences means that there are $C,h>0$ such that
$M_p\leq Ch^{p}N_{p},$ $p\in\mathbb{N}$. The stronger relation $M_p\prec N_p$ means that the latter inequality remains valid for every $h>0$ and a suitable $C=C_{h}>0$.

Let $\Omega \subseteq \mathbb{R}^d$ be open. For $K \Subset \Omega$ and $h > 0$, one writes $\mathcal{E}^{\{M_p\},h}(K)$ for the space of all $\varphi \in C^\infty(\Omega)$ such that
\[ \| \varphi \|_{\mathcal{E}^{\{M_p\},h}(K)} := \sup_{\substack{x \in K \\ \alpha \in \mathbb{N}^d}} \frac{|\varphi^{(\alpha)}(x)|}{h^{|\alpha|}M_{|\alpha|}} < \infty, \]
and $\mathcal{D}^{\{M_p\},h}_{K}$ stands for the closed subspace of $\mathcal{E}^{\{M_p\},h}(K)$ consisting of functions with compact support in $K$. Further on,
\[ \mathcal{D}^{(M_p)}(\Omega) = \varinjlim_{K \Subset \Omega} \varprojlim_{h \rightarrow 0^+} \mathcal{D}^{\{M_p\},h}_{K}, \qquad \mathcal{D}^{\{M_p\}}(\Omega) = \varinjlim_{K \Subset \Omega} \varinjlim_{h \rightarrow \infty} \mathcal{D}^{\{M_p\},h}_{K}, \]

and
\[ \mathcal{E}^{(M_p)}(\Omega) = \varprojlim_{K \Subset \Omega} \varprojlim_{h \rightarrow 0^+} \mathcal{E}^{\{M_p\},h}(K), \qquad \mathcal{E}^{\{M_p\}}(\Omega) = \varprojlim_{K \Subset \Omega} \varinjlim_{h \rightarrow \infty} \mathcal{E}^{\{M_p\},h}(K);  \]
their duals are the spaces of $M_p$-ultradistributions and compactly supported $M_p$-ultradistributions of Beurling and Roumieu type, respectively \cite{Komatsu}.

It is important to point out that under certain circumstances these spaces coincide with those discussed in Subsection \ref{subsection weights}. For instance, Petzsche and Vogt have shown \cite[Sect. 5]{Petzsche-V} (see also \cite[Satz 2.3]{Petzsche78}) that if the weight sequence $(M_p)_{p}$ satisfies $(M.1)$, $(M.2)$, $(M.3)'$, and the condition
\begin{itemize}
\item [$(M.4)\:$] $\displaystyle \frac{M_{p}}{p!}\subset \left(\frac{M_p}{p!}\right)^{c}$,
\end{itemize}
then one can always find a weight function $\omega$ fulfilling $(\alpha)$, $(\beta)$, $(\gamma_{0})$, and $\omega\asymp M$,
such that $\mathcal{D}_{(\omega)}(\Omega)=\mathcal{D}^{(M_p)}(\Omega)$ and $\mathcal{D}_{\{\omega\}}(\Omega)=\mathcal{D}^{\{M_p\}}(\Omega)$, topologically. Furthermore, they proved, under $(M.2)$, that $(M.4)$ is equivalent to the so-called Rudin condition:
\begin{itemize}
\item [$(M.4)''$] $\displaystyle \max_{q\leq p}\left(\frac{M_{q}}{q!}\right)^{\frac{1}{q}}\leq A\left(\frac{M_{p}}{p!}\right)^{\frac{1}{p}},$ $p\in\mathbb{N}$, for some $A>0$.
\end{itemize}
($(M.4)''$ is equivalent to the property that $\mathcal{E}^{(M_p)}(\Omega)$ and $\mathcal{E}^{\{M_p\}}(\Omega)$ are inverse closed, cf. \cite{Rudin62,Rainer-S}.) Finally, it is worth mentioning that strong non-quasianalyticity (i.e., Komatsu's condition (M.3) \cite{Komatsu}) automatically yields $(M.4)$, as shown by Petzsche \cite[Prop. 1.1]{Petzsche88}.

\section{Periodic ultradistributions}
\label{periodic ultradistributions}
 Let $\Lambda$ be a lattice in $\mathbb{R}^{d}$. An ultradistribution $f$ is said to be $\Lambda$-periodic if $f(\:\cdot\:+\mu)=f$ for all $\mu\in\Lambda$. We denote as $\mathcal{D}'_{\mathfrak{p}_{\Lambda},(\omega)}$ and $\mathcal{D}'_{\mathfrak{p}_{\Lambda},\{\omega\}}$ the spaces of $\Lambda$-periodic $\omega$-ultradistributions of Beurling and Roumieu type, respectively. We also consider $\mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)}=\mathcal{E}_{(\omega)}(\mathbb{R}^{d})\cap \mathcal{D}'_{\mathfrak{p}_{\Lambda},(\omega)}$ and $\mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}}=\mathcal{E}_{\{\omega\}}(\mathbb{R}^{d})\cap \mathcal{D}'_{\mathfrak{p}_{\Lambda},\{\omega\}}$, which are easily seen to be closed subspaces of $\mathcal{E}_{(\omega)}(\mathbb{R}^{d})$ and $\mathcal{E}_{\{\omega\}}(\mathbb{R}^{d})$, respectively.

We shall show in this section that every $\Lambda$-periodic $\omega$-ultradistribution can be expanded into a Fourier series. For it, we select a \emph{fundamental region} $I_{\Lambda}$ for the lattice, namely, a connected set $I_{\Lambda}\subseteq \mathbb{R}^{d}$ with the property that the restriction of the quotient mapping $\mathbb{R}^{d}\to \mathbb{R}^{d}/\Lambda$ to $I_{\Lambda}$  is a bijection. The set  $I_{\Lambda}$ is of course a $d$-dimensional parallelepiped and $\mathbb{R}^{d}=\bigcup_{\mu\in\Lambda}(\mu+I_\Lambda)$. It can be readily shown that $|\Lambda|:=\operatorname*{vol}(I_{\Lambda})$ does not depend on the choice of the fundamental region $I_{\Lambda}$. As in the introduction, $\Lambda^{\ast}$ stands for the dual lattice of $\Lambda$. Clearly, $\Lambda=\Lambda^{\ast\ast}$. Many of our arguments in this section are based on the Poisson summation formula \cite{Hormander}, which in this context takes the form
\begin{equation}
\label{Poisson}
\sum_{\mu\in\Lambda}e^{2\pi i\xi \cdot \mu}=\frac{1}{|\Lambda|}\sum_{\mu^*\in\Lambda^*}\delta(\xi-\mu^*) \quad \mbox{in }\mathcal{S}'(\mathbb{R}^d).
\end{equation}
Given $f\in\mathcal{E}'_{(\omega)}(\mathbb{R}^{d})$ (or whenever it makes sense), we denote as $f_{\mathfrak{p}_{\Lambda}}$ its \emph{$\Lambda$-periodization}, that is, the $\Lambda$-periodic $\omega$-ultradistribution
\begin{equation}
\label{eqperiodization}
f_{\mathfrak{p}_{\Lambda}}:=\sum_{\mu\in\Lambda}f(\:\cdot\:+\mu).
\end{equation}
We begin with a useful lemma.

\begin{lemma}\label{lemma partition unity}
There is $\eta\in\mathcal{D}_{(\omega)}(\mathbb{R}^{d})$ such that $\eta_{\mathfrak{p}_{\Lambda}}=1$, namely,
\begin{equation}
\label{eqpu}
\sum_{\mu\in\Lambda}\eta(x+\mu)=1, \quad x\in\mathbb{R}^{d}.
\end{equation}
\end{lemma}
\begin{proof} Fourier transforming (\ref{eqpu}) and employing the Poisson summation formula (\ref{Poisson}), one obtains that the relation (\ref{eqpu}) would be satisfied if we find $\eta\in\mathcal{D}_{(\omega)}(\mathbb{R}^{d})$ such that $\widehat{\eta}(0)=|\Lambda|$ and $\widehat{\eta}(\mu^*)=0$ for every $\mu^*\in\Lambda^*\setminus\{0\}$. Select $d$ vectors $\mu_{1},\dots,\mu_{d}$ that generate the Abelian group $\Lambda$ and that span $\mathbb{R}^{d}$, and pick $\varphi\in\mathcal{D}_{(\omega)}(\mathbb{R}^{d})$ such that $\int_{\mathbb{R}^{d}}\varphi(x)\mathrm{d}x=|\Lambda|$. The function $\eta$ given in Fourier side as
$$\widehat{\eta}(\xi)=\widehat{\varphi}(\xi)\prod_{j=1}^{d}\frac{\sin(2\pi\xi \cdot  \mu_j)}{2\pi \xi \cdot  \mu_j}$$
 satisfies all requirements.
\end{proof}

We employ the notation $e_y(x) = e^{2\pi i y \cdot x}$, where $y$ is a fixed vector of $\mathbb{R}^d$. Given a locally integrable $\Lambda$-periodic function $\phi$, its Fourier coefficients with respect to the lattice $\Lambda$ are given by
\begin{equation}
\label{eqfcoeff} c_{\mu^*}=c_{\mu^*}(\phi)=\frac{1}{|\Lambda|}\int_{I_{\Lambda}}\phi(x)e_{-\mu^*}(x)\mathrm{d}x, \quad \mu^*\in\Lambda^*.
\end{equation}
Obviously, the integral in (\ref{eqfcoeff}) does not depend on the choice of $I_{\Lambda}$; in fact, these coefficients can be also computed as
\begin{equation}
\label{eqfcoeff2} c_{\mu^*}=\frac{1}{|\Lambda|}\int_{\mathbb{R}^{d}}\phi(x)\eta(x)e_{-\mu^*}(x)\mathrm{d}x=\frac{1}{|\Lambda|}\widehat{(\phi\eta)}(\mu^*),
 \end{equation}
where $\eta$ is as in Lemma \ref{lemma partition unity}. Note that we thus have $c_{\mu^*}(e_{\nu^*})=\delta_{\nu^{*}\mu^{*}}$, as  immediately follows from (\ref{eqfcoeff2}). The next lemma shows that $\mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)}$ and $\mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}}$ are isomorphic to t.v.s of $\omega$-rapidly decreasing functions on the dual lattice $\Lambda^{\ast}$, namely,
$$\mathcal{S}_{(\omega)}(\Lambda^{\ast})=\varprojlim_{\lambda\to\infty} \mathcal{S}^{\lambda}_{\omega}(\Lambda^{\ast})\quad\mbox{and}\quad \mathcal{S}_{\{\omega\}}(\Lambda^{\ast})=\varinjlim_{\lambda\to0^{+}} \mathcal{S}^{\lambda}_{\omega}(\Lambda^{\ast}),$$
where
$$\displaystyle\mathcal{S}^{\lambda}_{\omega}(\Lambda^{\ast})=\left\{(c_{\mu^*})_{\mu^*\in\Lambda^*}\in \mathbb{C}^{\Lambda^{\ast}}:\: \sigma_{\lambda}((c_{\mu^*})):= \sup_{\mu^*\in\Lambda^*}|c_{\mu^*}|e^{\lambda \omega(\mu^*)}<\infty \right\}, \quad \lambda\in\mathbb{R}.$$
(In the Roumieu case we assume that $(\gamma_{0})$ holds.)
\begin{lemma}\label{ultrafunc}
If $\phi \in\mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)}$ $($$\phi \in\mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}}$$)$, then
\begin{equation}
\phi= \sum_{\mu^* \in \Lambda^\ast} c_{\mu^*}e_{\mu^*},
\label{series}
\end{equation}
with convergence in $\mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)}$ $($$\mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}}$$)$,
where the Fourier coefficients $c_{\mu^*}$ are given by $(\ref{eqfcoeff})$. Moreover, the mapping $\phi\mapsto (c_{\mu^*})_{\mu^*\in\Lambda^*}$ yields the $($t.v.s.$)$ isomorphisms $\mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)} \cong \mathcal{S}_{(\omega)}(\Lambda^{\ast})$ and $\mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}} \cong \mathcal{S}_{\{\omega\}}(\Lambda^{\ast})$.
\end{lemma}
\begin{proof} Let $\eta\in\mathcal{D}_{(\omega)}(\mathbb{R}^{d})$ be as in Lemma \ref{lemma partition unity}. The relation (\ref{eqfcoeff2}) then yields
$$
 |c_{\mu^*}(\phi)| =|\Lambda|^{-1} \left|\widehat{\eta \phi}(\mu^*)\right| \leq |\Lambda|^{-1}\|\eta \phi\|_\lambda e^{-\lambda\omega(\mu^*)},
 $$
which shows the continuity of $\phi\mapsto (c_{\mu^*}(\phi))_{\mu^*\in\Lambda^*}$ in both cases. On the other hand, if $(c_{\mu^*})_{\mu^*\in\Lambda^*}\in \mathcal{S}_{(\omega)}(\Lambda^{\ast})$ (or $(c_{\mu^*})_{\mu^*\in\Lambda^*}\in \mathcal{S}_{\{\omega\}}(\Lambda^{\ast})$), $F$ is a finite subset of $\Lambda^*$, and $\psi\in \mathcal{D}_{(\omega)}(\mathbb{R}^{d})$ ($\psi\in \mathcal{D}_{\{\omega\}}(\mathbb{R}^{d})$), then
$$
\| \psi \sum_{\mu^*\in\Lambda^*\setminus F  }c_{\mu^*}e_{\mu^*}\|_{\lambda}\leq \|\psi\|_{\lambda} \sigma_{2\lambda}((c_{\mu^*}))\sum_{\mu^*\in\Lambda^*\setminus F  } e^{-\lambda \omega(\mu^*)}\:.
$$
This proves that $\sum_{\mu^*\in\Lambda^* }c_{\mu^*}e_{\mu^*}$ is summable in $\mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)}$ (in $\mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}}$) and that the mapping $(c_{\mu^*})_{\mu^*\in\Lambda^*}\mapsto \sum_{\mu^*\in\Lambda^*}c_{\mu^*}e_{\mu*}$ is also continuous. The convergence of (\ref{series}) is now a consequence of the injectivity of $\phi\mapsto (c_{\mu^*}(\phi))_{\mu^*\in\Lambda^*}$, which of course follows from the case $\Lambda=\mathbb{Z}^{d}$ by a linear change of variables\footnote{Every lattice in $\mathbb{R}^{d}$ is of the form $\Lambda=T(\mathbb{Z}^{d})$, where $T$ is an invertible matrix.}. Alternatively, this injectivity can also be established as follows. If  $\phi$ is such that $c_{\mu^*}(\phi)=|\Lambda|^{-1}\widehat{(\phi \eta)}(\mu^{*})=0$ for every $\mu^*\in\Lambda^*$, then, for an arbitrary $\psi\in\mathcal{D}(\mathbb{R}^{d})$, we have $\int_{\mathbb{R}^{d}}\phi(x)\psi(x)\mathrm{d}x =\int_{\mathbb{R}^{d}}\phi(x)\eta(x)\psi_{\mathfrak{p}_{\Lambda}}(x)
 \mathrm{d}x$. Hence, by the Poisson summation formula,
\begin{align*}
\int_{\mathbb{R}^{d}}\phi(x)\psi(x)\mathrm{d}x&=\frac{1}{|\Lambda|}\left\langle \widehat{(\phi \eta)}(-\xi),\sum_{\mu^*\in\Lambda^*}\widehat{\psi}(\xi)\delta(\xi-\mu^*)\right\rangle
\\
&
=\frac{1}{|\Lambda|}\sum_{\mu^*\in\Lambda^*}\widehat{\psi}(-\mu^*)\widehat{(\phi \eta)}(\mu^{*})=0.
\end{align*}
Therefore, we must have $\phi=0$.
\end{proof}

We are ready to deal with Fourier expansions of $\omega$-ultradistributions. The treatment is similar to the distribution case \cite{zeeman}, but we give the details for the sake of completeness. Observe first that a standard argument shows that $$\mathcal{S}'_{(\omega)}(\Lambda^{\ast})= \varinjlim_{\lambda\to\infty} \mathcal{S}^{-\lambda}_{\omega}(\Lambda^{\ast})\quad\mbox{and}\quad \mathcal{S}'_{\{\omega\}}(\Lambda^{\ast})=\varprojlim_{\lambda\to0^{+}}\mathcal{S}^{-\lambda}_{\omega}(\Lambda^{\ast}).$$
Lemma \ref{ultrafunc} then yields that the duals of $\mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)}$ and $\mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}}$ are isomorphic to these spaces and their elements can be expanded as $g=\sum_{\mu^* \in \Lambda^{\ast}}c_{\mu^*}(g)e_{\mu^\ast}$ with $( c_{\mu^*}(g))_{\mu^*\in\Lambda^*}$ being an element of $\mathcal{S}'_{(\omega)}(\Lambda^{\ast})$ or $\mathcal{S}'_{\{\omega\}}(\Lambda^{\ast})$, respectively, and
$$
\langle g,\phi \rangle_{(\mathcal{D}_{\mathfrak{p}_{\Lambda},\omega})'\times \mathcal{D}_{\mathfrak{p}_{\Lambda},\omega}}=|\Lambda| \sum_{\mu^*\in\Lambda^*} c_{-\mu^*}(g)c_{\mu^*}(\phi),
$$
where $\omega$ stands for either the Beurling case $(\omega)$ or the Roumieu case $\{\omega\}$, respectively.

In the rest of the discussion $\eta$ stands for a test function as in Lemma \ref{lemma partition unity}. Given
$f\in \mathcal{D}'_{\mathfrak{p}_{\Lambda},(\omega)}$ ($f\in\mathcal{D}'_{\mathfrak{p}_{\Lambda},\{\omega\}}$), we identify it with an element $g$ of the dual of  $\mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)}$ ($\mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}}$) as follows:
\begin{equation}
\label{eqidentificationperiodic}
\langle g,\phi \rangle_{(\mathcal{D}_{\mathfrak{p}_{\Lambda},\omega})'\times \mathcal{D}_{\mathfrak{p}_{\Lambda},\omega}}:=\langle f, \eta \phi \rangle_{\mathcal{D}'_{\omega}(\mathbb{R}^{d})\times \mathcal{D}_{\omega}(\mathbb{R}^{d})}, \quad \phi\in \mathcal{D}_{\mathfrak{p}_{\Lambda},\omega}\:.
\end{equation}
The definition of $g$ is independent of the choice of $\eta$, as can be readily verified. The $\Lambda$-periodic $\omega$-ultradistribution $f$ can be recovered on $\mathbb{R}^{d}$ from $g$ as
\begin{equation}
\label{eqidentificationperiodic2}
\langle f,\varphi\rangle_{\mathcal{D}'_{\omega}(\mathbb{R}^{d})\times \mathcal{D}_{\omega}(\mathbb{R}^{d})}=\langle g,\varphi_{\mathfrak{p}_{\Lambda}} \rangle_{(\mathcal{D}_{\mathfrak{p}_{\Lambda},\omega})'\times \mathcal{D}_{\mathfrak{p}_{\Lambda,\omega}}}\:, \quad \varphi\in  \mathcal{D}_{\omega}(\mathbb{R}^{d}).
\end{equation}
Using $g$, one defines the Fourier coefficients of $f$ as
\begin{equation}
\label{eqfcoeff3}
c_{\mu^*}(f):=c_{\mu^*}(g)=\frac{1}{|\Lambda|}\langle f, \eta e_{-\mu^*} \rangle, \quad \mu^*\in\Lambda^*.
\end{equation}
Summarizing, we have:
\begin{proposition}\label{perdis} Every $f\in \mathcal{D}'_{\mathfrak{p}_{\Lambda},(\omega)}$ $($$f\in\mathcal{D}'_{\mathfrak{p}_{\Lambda},\{\omega\}}$$)$ can be expanded as
\begin{equation}
\label{series2}
f=\sum_{\mu^* \in \Lambda^*} c_{\mu^*}e_{\mu^*} \quad \mbox{in } \mathcal{D}'_{(\omega)}(\mathbb{R}^{d}) \quad (\mbox{in } \mathcal{D}'_{\{\omega\}}(\mathbb{R}^{d})),
\end{equation}
where the Fourier coefficients $c_{\mu^*}$ are given by $(\ref{eqfcoeff3})$. The mapping $f\mapsto (c_{\mu^*})_{\mu^*\in\Lambda^*}$ provides the $($t.v.s.$)$ isomorphisms $\mathcal{D}'_{\mathfrak{p}_{\Lambda},(\omega)} \cong \mathcal{S}'_{(\omega)}(\Lambda^{\ast})$ and $\mathcal{D}'_{\mathfrak{p}_{\Lambda},\{\omega\}} \cong \mathcal{S}'_{\{\omega\}}(\Lambda^{\ast})$.
\end{proposition}
\begin{proof} Writing  $g=\sum_{\mu^* \in \Lambda^*} c_{\mu^*}(f)e_{\mu^*}$ for the ultradistribution given by (\ref{eqidentificationperiodic}), we obtain
$$\langle f,\varphi\rangle=\langle g,\varphi_{\mathfrak{p}_{\Lambda}} \rangle=|\Lambda|\sum_{\mu^* \in \Lambda^*} c_{\mu^*}(f) c_{-\mu^*}(\varphi_{\mathfrak{p}_{\Lambda^*}} )=\sum_{\mu^* \in \Lambda^*} c_{\mu^*}(f)\int_{\mathbb{R}^{d}}\eta(x)\varphi_{\mathfrak{p}_{\Lambda}}(x)e_{\mu^*}(x) \mathrm{d}x,
$$ which proves (\ref{series2}) because $\int_{\mathbb{R}^{d}}\varphi(x)e_{\mu^*}(x)\mathrm{d}x =\int_{\mathbb{R}^{d}}\eta(x)\varphi_{\mathfrak{p}_{\Lambda}}(x)e_{\mu^*}(x)\mathrm{d}x $. Note that the correspondence $f\mapsto g$ provides continuous mappings $\mathcal{D}'_{\mathfrak{p}_{\Lambda},(\omega)}\to (\mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)})'$ and $\mathcal{D}'_{\mathfrak{p}_{\Lambda},\{\omega\}}\to (\mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}})'$,  as directly follows from (\ref{eqidentificationperiodic}). Their inverse mappings (cf. (\ref{eqidentificationperiodic2})) are the transposes of the continuous mappings $\mathcal{D}_{(\omega)}(\mathbb{R}^{d})\to \mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)}$ and $\mathcal{D}_{\{\omega\}}(\mathbb{R}^{d})\to \mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}}$ given by $\varphi\mapsto\varphi_{\mathfrak{p}_{\Lambda}}$, so they are continuous as well.
\end{proof}

In view of Proposition \ref{perdis}, we can canonically identify  $\mathcal{D}'_{\mathfrak{p}_{\Lambda},(\omega)} $ and $\mathcal{D}'_{\mathfrak{p}_{\Lambda},\{\omega\}}$ with the duals of $\mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)}$ and $\mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}}$ via the Fourier series (\ref{series2}), so that we simply write $\mathcal{D}'_{\mathfrak{p}_{\Lambda},(\omega)}=(\mathcal{D}_{\mathfrak{p}_{\Lambda},(\omega)})' $ and $\mathcal{D}'_{\mathfrak{p}_{\Lambda},\{\omega\}}=(\mathcal{D}_{\mathfrak{p}_{\Lambda},\{\omega\}})'$. This convention of course amounts to the same as the identification $f=g$ by means of (\ref{eqidentificationperiodic}) and
(\ref{eqidentificationperiodic2}).

We can also define weighted Fourier-Lebesgue spaces with respect to the dual lattice $\Lambda^{*}$. Let $v \in \mathscr{M}_{(\omega)}$ and $q \in[1, \infty]$. The Banach space $\mathcal{F} l^q_{v,\Lambda^{*}}$ consists of all $f =\sum_{\mu^*\in\Lambda^*} c_{\mu^*}e_{\mu^*}\in \mathcal{D}'_{\mathfrak{p}_{\Lambda},(\omega)}$ such that $(c_{\mu^*})_{\mu^* \in \Lambda^*} \in l^q_v(\Lambda^*)$, namely,
$$
|f|_{\mathcal{F} l^{q}_{v,\Lambda^*}} := \|(c_{\mu^*})_{\mu^*\in\Lambda^*}\|_{l^q_v(\Lambda^*)}=\|(c_{\mu^*}v(\mu^*))_{\mu^*\in\Lambda^*}\|_{l^q(\Lambda^*)} < \infty.
$$

\begin{remark}
\label{rFSu} Analogous results hold for $M_p$-ultradistributions under the assumptions $(M.1)$, $(M.2)'$, and $(M.3)'$. In fact, assume that the sequence $(M_{p})_{p\in\mathbb{N}}$ satisfies these three conditions  and consider ${\mathcal{D}^{(M_p)}_{\mathfrak{p}_{\Lambda}}}'$ and ${\mathcal{D}^{\{M_p\}}_{\mathfrak{p}_{\Lambda}}}'$, the subspaces of ${\mathcal{D}^{(M_p)}}'(\mathbb{R}^d)$ and ${\mathcal{D}^{\{M_p\}}}'(\mathbb{R}^d)$, respectively, consisting of $\Lambda$-periodic $M_p$-ultradistributions. Then, every $\Lambda$-periodic $M_p$-ultradistribution admits the Fourier expansion (\ref{series2}) with convergence in ${\mathcal{D}^{(M_p)}}'(\mathbb{R}^d)$ or ${\mathcal{D}^{\{M_p\}}}'(\mathbb{R}^d)$, respectively. Furthermore,  $f\mapsto (c_{\mu^*})_{\mu^*\in\Lambda^*}$ also yields t.v.s. isomorphisms $\displaystyle{\mathcal{D}^{(M_p)}_{\mathfrak{p}_{\Lambda}}}' \cong {\mathcal{S}^{(M_p)}}'(\Lambda^{\ast}):= \varinjlim_{h\to\infty} \mathcal{S}^{M_p,-h}(\Lambda^{\ast})$ and $\displaystyle{\mathcal{D}^{\{
 M_p\}}_{
 \mathfrak{p}_{\Lambda}}}' \cong {\mathcal{S}^{\{M_p\}}}'(\Lambda^{\ast}):=\varprojlim_{h\to0^{+}}\mathcal{S}^{M_p,-h}(\Lambda^{\ast})$, where
$$
\mathcal{S}^{M_p,-h}(\Lambda^{\ast})=\left\{(c_{\mu^*})_{\mu^*\in\Lambda^*}\in \mathbb{C}^{\Lambda^{\ast}}:\: \sup_{\mu^*\in\Lambda^*}|c_{\mu^*}|e^{- M(h\mu^*)}<\infty\right\}.
$$
The proofs of these assertions can be obtained exactly as for $\omega$-ultradistributions. We also refer to \cite{Gorba,Petzsche78} for studies involving Fourier series of $M_p$-ultradistributions.

\end{remark}
\smallskip

\section{Wave front sets of Fourier-Lebesgue type}
\label{Section F-L}
The aim of this section is to provide a discrete characterization of wave front sets of Fourier-Lebesgue type. Besides the conditions $(\alpha)$, $(\beta)$, and $(\gamma)$, we impose throughout this section the following additional assumption on $\omega$: The weight $\omega$ is a non-decreasing
function of $|\xi|$, namely,
\begin{itemize}
\item [$(\alpha_0)$] $\omega(\xi)=\omega_{0}(|\xi|)$, $\xi\in\mathbb{R}^{d}$, where $\omega_{0}:[0,\infty)\to[0,\infty)$ is non-decreasing.
\end{itemize}

We need to introduce some notation in order to define the Fourier-Lebesgue wave front set of an $\omega$-ultradistribution.
 Let $v\in\mathscr{M}_{(\omega)}$, let $q\in[1,\infty]$, and let $\Gamma$ be a cone in $\mathbb{R}^d$. If $g\in\mathcal{S}'_{(\omega)}(\mathbb{R}^d)$ is such that $\widehat{g}$ is locally integrable in an open neighborhood of $\Gamma$, we consider the seminorm
\begin{equation}
\label{eqsFL}
|g|_{\mathcal{F} L^{q,\Gamma}_v} :=\|\widehat{g}\|_{L^q_{v}(\Gamma)}=\|\widehat{g}\:1_\Gamma \|_{L^q_{v}},
\end{equation}
where $1_A$ stands for the characteristic function of a set $A$. The seminorm (\ref{eqsFL}) is in particular well-defined if $g\in\mathcal{E}'_{(\omega)}(\mathbb{R}^d)$, but naturally it might become $\infty$.

Let now $f\in\mathcal{D}'_{(\omega)}(\Omega)$. The $\omega$-ultradistribution $f$ is said to be \emph{$\mathcal{F} L^q_v$-microlocally regular} at the point $(x_0,\xi_0)\in \Omega\times (\mathbb{R}^d \setminus \{ 0 \})$ if there are an open conic neighborhood $\Gamma$ of $\xi_0$ and a test function $\varphi\in\mathcal{D}_{(\omega)}(\Omega)$ with $\varphi(x_0)\neq0$ such that
\begin{equation}
\label{eqe1}
|\varphi f|_{\mathcal{F} L^{q,\Gamma}_v }<\infty.
\end{equation}
The wave front set $WF_{\mathcal{F} L^q_v}(f)$ consists of all those points $(x_0,\xi_0)\in \Omega\times (\mathbb{R}^d \setminus \{ 0 \})$ such that $f$ is \emph{not} $\mathcal{F} L^q_v$-microlocally regular at  $(x_0,\xi_0)$.

 We are now ready to state the main theorem of this section, a discrete characterization of $WF_{\mathcal{F} L^q_v}(f)$ with respect to a lattice.

\begin{theorem}\label{main} Let $\Lambda$ be a lattice in $\mathbb{R}^{d}$, $f \in \mathcal{D}_{(\omega)}'(\Omega)$, $v\in\mathscr{M}_{(\omega)}$, $q\in[1,\infty]$, and $(x_0, \xi_0) \in \Omega \times (\mathbb{R}^d \backslash \{ 0 \})$. Suppose that $U$ is an  open convex neighborhood of the origin such that $U\cap\Lambda^{*}=\{0\}$ and $x_0+U\subseteq\Omega$. Then, the following statements are equivalent:
\begin{itemize}
\item[$(i)$] There are an open conic neighborhood $\Gamma$ of $\xi_0$ and $\varphi \in \mathcal{D}_{(\omega)}(x_{0}+U)$ with $\varphi(x_0)\neq 0$ such that
\[
 \left\|(\widehat{\varphi f}(\mu))_{\mu\in\Gamma\cap\Lambda} \right\|_{l^{q}_{v}(\Gamma\cap\Lambda)} < \infty.
\]\item[$(ii)$] $f$ is $\mathcal{F} L^q_v$-microlocally regular at  $(x_0,\xi_0)$.
\end{itemize}
\end{theorem}
Part of the proof of Theorem \ref{main} is based on the ensuing lemma:

\begin{lemma}\label{mainlemma} $\mbox{ }$
\begin{itemize}
\item[$(i)'$] Condition $(i)$ from Theorem \ref{main} implies that there are an open conic neighborhood $\Gamma_1$ of $\xi_0$ and an open neighborhood $U_1\subseteq U$ of the origin such that for every bounded set $B \subseteq \mathcal{D}_{(\omega)}(x_0+U_1)$
\[ \sup_{\psi \in B} \left\|(\widehat{\psi f}(\mu))_{\mu\in \Gamma_1\cap\Lambda} \right\|_{l^{q}_{v}(\Gamma_1\cap\Lambda)}<\infty .\]
\item[$(ii)'$] Condition $(ii)$ from Theorem \ref{main} implies that there are an open conic neighborhood $\Gamma_1$ of $\xi_0$ and an open neighborhood $U_1$ of the origin such that for every bounded set $B \subseteq \mathcal{D}_{(\omega)}(x_0+U_1)$
\[ \sup_{\psi \in B} |\psi f|_{ \mathcal{F}L^{q,\Gamma_1}_{v}}
< \infty.\]
\end{itemize}
\end{lemma}
\begin{proof}
We only prove $(i)'$, because the second assertion can be established in a similar fashion by replacing sums by integrals. Assume condition $(i)$ of Theorem \ref{main}. We begin by finding suitable $\Gamma_1$ and $U_1$.
Choose an open conic neighborhood $\Gamma_1$ of $\xi_0$ such that $\overline{\Gamma}_1 \subseteq \Gamma \cup \{ 0 \}$. Let $m \in \mathbb{Z}_+$  be such that $m^{-1}$ is smaller than the distance between $\partial \Gamma$ and the intersection of $\Gamma_1$ with the unit sphere, and also smaller than
the distance between  $\partial \Gamma_1$  and the intersection of $\overline{\mathbb{R}^d \backslash \Gamma}$ with the unit
sphere. Hence $\xi \in \Gamma_1$ and $y \notin \Gamma$ imply that $|\xi - y| \geq  m^{-1}\max(|\xi|, |y|)$. As $U_1$ we select any open neighborhood $U_1$ of $0$ such that $\overline{U}_1\Subset U\cap\{x:\:\varphi(x+x_0)\neq 0\}=:U_2$.  In addition, pick $\kappa\in\mathcal{D}_{(\omega)}(x_0+U_2)$ such that $\kappa\equiv 1$ on $x_0+U_1$.  Observe that\footnote{In fact, Beurling theorem \cite{beurling,Bjorck} tells us that the condition $(\beta)$ is equivalent to the regularity of the Beurling algebras $L^{1}_{e^{\omega_{\lambda}}}$ with weights $\omega_{\lambda}(\xi):=\lambda\omega(\xi)$. As a consequence of the general theory of regular commutative Banach algebras \cite{G-R-S}, one obtains that analytic functions act locally on each $\mathcal{F}(L^{1}_{e^{\omega_{\lambda}}})$; in particular, $\mathcal{E}_{(\omega)}(\mathbb{R}^{d})$ and $\mathcal{E}_{\{\omega\}}(\mathbb{R}^{d})$ are inverse closed.} $\theta:=\kappa/\varphi \in\mathcal{D}_{(\omega)}(x_0+U_2)$. Let now
 $B\subseteq \mathcal{D}_
 {(\omega)}(x_0+U_1)$ be a bounded subset. If $\psi\in B$ we have that $\psi f=\varphi \psi_1 f$, where $\psi_1$ is an element of the bounded set $B_1:=\theta B\subseteq \mathcal{D}_{(\omega)}(x_0+U_1)$.  Next, notice that there is a fundamental region $I_{\Lambda^{\ast}}$ of the dual lattice $\Lambda^*$ such that $x_0+U\subseteq I_{\Lambda^*}$. Since for an arbitrary $g\in \mathcal{E}'_{(\omega)}(\mathbb{R}^{d})$ having support in  the interior of $I_{\Lambda^{\ast}}$ we have that the Fourier coefficients of its $\Lambda^*$-periodization (cf. Section \ref{periodic ultradistributions}) are given by $(\widehat{g}(\mu))_{\mu\in\Lambda}$, we obtain
\[ (\varphi f)_{p_{\Lambda^*}} = \sum_{\mu \in \Lambda} a_\mu e_\mu, \]
with $a_\mu=\widehat{\varphi f}(\mu)$. Furthermore, since $(\psi f)_{\mathfrak{p}_{\Lambda^*}}= (\varphi f)_{\mathfrak{p}_{\Lambda^*}}(\psi_1)_{\mathfrak{p}_{\Lambda^*}}$, we conclude that
\[ \widehat{\psi f}(\mu) = \sum_{\beta \in \Lambda} a_\beta \widehat{\psi}_1(\mu-\beta), \qquad \mu\in\Lambda.  \]
Thus, by (\ref{moderate}),
\begin{align*}
 \left\|(\widehat{\psi f}(\mu))_{\mu\in\Gamma_1\cap \Lambda} \right\|_{l^{q}_{v}(\Gamma_1\cap\Lambda)}
&\leq C  \left \|\left(\sum_{\beta\in\Lambda}|a_\beta|v(\beta)|\widehat{\psi_1}(\mu-\beta)|e^{\lambda \omega(\mu-\beta)} \right)_{\mu\in\Gamma_1\cap\Lambda}\right \|_{l^q(\Gamma_1\cap\Lambda)}
 \\
& \leq C(I_1(\psi_1) + I_2(\psi_1)), \quad \forall\psi\in B,
\end{align*}
where
\[ I_1(\psi_1) = \left \|\left(\sum_{\beta  \in \Gamma \cap \Lambda}|a_\beta|v(\beta)|\widehat{\psi_1}(\mu-\beta)|e^{\lambda \omega(\mu-\beta)}\right)_{\mu\in\Gamma_1\cap\Lambda} \right \|_{l^q(\Gamma_1\cap\Lambda)},
\]
and
\[ I_2(\psi_1) = \left \|\left(\sum_{\beta  \notin \Gamma \cap \Lambda}|a_\beta|v(\beta)|\widehat{\psi_1}(\mu-\beta)|e^{\lambda \omega(\mu-\beta)} \right)_{\mu\in\Gamma_1\cap\Lambda}\right \|_{l^q(\Gamma_1\cap\Lambda)}
.\]
Young's inequality and the boundedness of $B_1$ imply that
\[ \sup_{\psi_1 \in B_1}  I_1(\psi_1) \leq  \left\|(\widehat{\varphi f}(\mu))_{\mu\in\Gamma\cap \Lambda} \right\|_{l^{q}_{v}(\Gamma\cap\Lambda)} \sup_{\psi_1 \in B_1} \sum_{\beta \in \Lambda} |\widehat{\psi_1}(\beta)|e^{\lambda\omega(\beta)} < \infty. \]
We now estimate $I_2(\psi_1)$. By Proposition \ref{perdis} and the fact that $v$ is $\omega$-moderate there exist $D, \lambda_0 > 0$ such that
\begin{equation}
\label{eqe2}
 |a_\beta|v(\beta) \leq De^{\lambda_0\omega(\beta)}, \qquad \forall \beta \in \Lambda.
\end{equation}
Since $B_1$ is bounded, we have that for every $\gamma > 0$ there exists $C_\gamma > 0$ such that
\begin{equation}
\label{eqe3}
 \sup_{\psi_1 \in B_1}|\widehat{\psi_1}(\xi)| \leq C_\gamma e^{-\gamma\omega(\xi)}, \qquad \forall \xi \in \mathbb{R}^d.
 \end{equation}
Hence
\[\sup_{\psi_1 \in B_1}  I_2(\psi_1) \leq A \left \|\left( \sum_{\beta \notin \Gamma \cap \Lambda}e^{\lambda_0\omega(\beta)-\gamma_0\omega(\mu-\beta)}\right)_{\mu\in\Gamma_1\cap\Lambda}\right \|_{l^q(\Gamma_1\cap\Lambda)}\]
where $A = DC_{\gamma_0 + \lambda}$. In view of the choice of the cone $\Gamma_1$ and the constant $m$, we have that
\begin{align*}
\sup_{\psi_1 \in B_1}  I_2(\psi_1) &\leq A \left \|\left(e^{-(\gamma_0/2)\omega(\mu/m)} \sum_{\beta  \notin \Gamma \cap \Lambda}e^{\lambda_0\omega(\beta)-(\gamma_0/2)\omega(\beta/m)}\right)_{\mu\in\Gamma_1\cap\Lambda}\right \|_{l^q(\Gamma_1\cap\Lambda)}
\\
&\leq A\left \|\left(e^{-(\gamma_0/2)\omega(\mu/m)}\right)_{\mu\in\Gamma_1\cap\Lambda}\right\|_{l^q(\Gamma_1\cap\Lambda)} \sum_{\beta \in \Lambda}e^{(\lambda_0m  -(\gamma_0/2))\omega(\beta/m)}  < \infty,
\end{align*}
provided that $\gamma_0$ is large enough.
\end{proof}

We can now proceed to show Theorem \ref{main}.

\begin{proof}[Proof of Theorem \ref{main}.] Let $I_{\Lambda}$ be a fundamental region for $\Lambda$ with $0\in I_{\Lambda}$.

$(i) \Rightarrow (ii)$: Let $\Gamma_1$ and $U_1$ be as in the first part of Lemma \ref{mainlemma}. Choose an open conic neighborhood $\Gamma_2$ of $\xi_0$ such that $\overline{\Gamma}_2 \subseteq \Gamma_1 \cup \{ 0 \}$ and $\psi \in \mathcal{D}_{(\omega)}(x_0+ U_1)$ with $\psi (x_0)\neq 0$.  Note that $B = \left \{ \psi_t := \psi e_{-t} \, : \, t \in I_{\Lambda}\right\}$ is a bounded subset of $\mathcal{D}_{(\omega)}(x_0+U_1)$. Fix $r > 0$ such
that $\Gamma_2 \cap \{ \xi \in \mathbb{R}^d \, : \, |\xi| \geq r \} \subseteq (\Gamma_1 \cap \Lambda) + I_{\Lambda}$.
Set $D = \sup_{t \in I_\Lambda}e^{\lambda \omega(t)}$. For $q < \infty$ the first part of Lemma \ref{mainlemma} implies that
\begin{align*}
\int_{\substack{ \xi \in \Gamma_2 \\ |\xi| \geq r }} |\widehat{\psi f}(\xi)v(\xi)|^q \mathrm{d}\xi & \leq \sum_{\mu\in \Gamma_1 \cap \Lambda} \int_{I_{\Lambda}}  |\widehat{\psi f}(t+\mu)v(t+\mu)|^q \mathrm{d}t \\
&\leq C^qD^q\sum_{\mu \in \Gamma_1 \cap \Lambda}v(\mu)^q \int_{I_{\Lambda}}  |\widehat{\psi_t f}(\mu)|^q \mathrm{d}t \\
&\leq|\Lambda| C^qD^q\sup_{t \in I_{\Lambda}}\sum_{\mu\in \Gamma_1 \cap \Lambda}|\widehat{\psi_t f}(\mu)v(\mu)|^q < \infty,
\end{align*}
while for $q=\infty$ we have
\begin{align*}
\underset{|\xi| \geq r}{\sup_{\xi \in \Gamma_2}} |\widehat{\psi f}(\xi)v(\xi)| & \leq \sup_{\mu\in \Gamma_1 \cap \Lambda} \sup_{t \in I_{\Lambda}}  |\widehat{\psi f}(t+\mu)v(t+\mu)| \\
&\leq CD\sup_{t \in I_{\Lambda}}\sup_{\mu \in \Gamma_1 \cap \Lambda}|\widehat{\psi_t f}(\mu)v(\mu)| < \infty.
\end{align*}
$(ii) \Rightarrow (i)$: The case $q=\infty$ is trivial, so we assume $q < \infty$. Let $\Gamma_1$ and $U_1$ be as in the second part of Lemma \ref{mainlemma} and let $\psi \in \mathcal{D}_{(\omega)}(x_0+ U_1)$ with $\psi (x_0)\neq 0$.  Choose an open conic neighborhood $\Gamma_2$ of $\xi_0$ such that $\overline{\Gamma}_2 \subseteq \Gamma_1 \cup \{ 0 \}$ and $ r > 0$ so large that
$(\Gamma_2 + I_\Lambda) \cap \{ \xi \in \mathbb{R}^d \, : \, |\xi| \geq r \} \subseteq \Gamma_1$. Hence
\begin{align*}
\left(\sum_{\mu \in \Gamma_2 \cap \Lambda} |\widehat{\psi f}(\mu)v(\mu)|^q \right)^{1/q} &= \left( \frac{1}{|\Lambda|}\sum_{\mu \in \Gamma_2 \cap \Lambda} \int_{\mu + I_\Lambda}|\widehat{\psi f}(\mu)v(\mu)|^q \mathrm{d}\xi \right)^{1/q} \\
&\leq |\Lambda|^{-1/q}(J_1^{1/q} + J_2^{1/q}),
\end{align*}
where
\[ J_1 := \sum_{\mu \in \Gamma_2 \cap \Lambda} \int_{\mu + I_\Lambda}|\widehat{\psi f}(\mu)-\widehat{\psi f}(\xi) |^qv(\mu)^q \mathrm{d}\xi, \]
and
\begin{align*}
 J_2 &:= \sum_{\mu \in \Gamma_2 \cap \Lambda} \int_{\mu + I_\Lambda}|\widehat{\psi f}(\xi) |^qv(\mu)^q \mathrm{d}\xi \\
&\leq \sum_{|\mu| \leq r} \int_{\mu + I_\Lambda}|\widehat{\psi f}(\xi) |^qv(\mu)^q \mathrm{d}\xi + C^q \sup_{t \in I_\Lambda} e^{q \lambda \omega(t)} \int_{\Gamma_1}|\widehat{\psi f}(\xi)v(\xi)|^q \mathrm{d}\xi < \infty.
\end{align*}
We now estimate $J_1$. Set $D = \sup_{t \in I_\Lambda} |t|$. Since $I_\Lambda$ is convex and contains the origin, we have for $\mu \in \Gamma_2 \cap \Lambda$ and $\xi \in \mu + I_\Lambda$ that
\begin{align*}
|\widehat{\psi f}(\mu)-\widehat{\psi f}(\xi) |^q &\leq |\mu - \xi|^q \sup_{t \in [0,1]} | \nabla \widehat{\psi f}(\xi + t(\mu-\xi) )|^q \\
&\leq D^q \sup_{y \in I_\Lambda} | \nabla \widehat{\psi f}(\xi - y )|^q   \\
&\leq D^q \sum_{k = 1}^d \sup_{y \in I_\Lambda} |  \mathcal{F}(x_k e_y\psi f)(\xi)|^q.
\end{align*}
Note that the set $\{ x_k e_y\psi: \:   y \in I_\Lambda\}$ is bounded in $\mathcal{D}_{(\omega)}(x_0 + U_1)$ for each $k = 1, \ldots, d$. Hence part $(ii)'$ of Lemma \ref{mainlemma} implies that
\begin{align*}
J_1 \leq &\sum_{|\mu| \leq r} \int_{\mu + I_\Lambda}|\widehat{\psi f}(\mu)-\widehat{\psi f}(\xi) |^qv(\mu)^q \mathrm{d}\xi \\
&+  C^qD^q \sup_{t \in I_\Lambda} e^{q\lambda \omega(t)} \sum_{k = 1}^d \sup_{y \in I_\Lambda}  \int_{\Gamma_1}|\mathcal{F}(x_k e_y\psi f)(\xi)|^q v(\xi)^q\mathrm{d}\xi < \infty,
\end{align*}
which concludes the proof of the theorem.
\end{proof}
\begin{remark}\label{rk4.3} Note that if the ultradistribution $f\in\mathcal{D}_{\{\omega\}}'(\Omega)$, then the regularity requirement on the $\varphi$ used in part $(i)$ from Theorem \ref{main} and in (\ref{eqe1}) for the definition of $\mathcal{F} L_{\nu}^{p}$-microregularity can be relaxed to: $\varphi \in \mathcal{D}_{\{\omega\}}(x_{0}+U)$ and $\varphi \in \mathcal{D}_{\{\omega\}}(\Omega)$, respectively, because the test function can always be replaced by one belonging to $\mathcal{D}_{(\omega)}(\Omega)$. In addition, if $v\in\mathscr{M}_{\{\omega\}}$, Lemma \ref{mainlemma} can be strengthened: The properties $(i)'$ and $(ii)'$  hold true for bounded subsets $B$ of $\mathcal{D}_{\{\omega\}}(x_0+U_1)$.
\end{remark}
\smallskip

The proof of Lemma \ref{mainlemma} motives the introduction of the following discrete seminorms for $\Lambda^{\ast}$-periodic $\omega$-ultradistributions. Let $\Gamma$ be a cone and $g=\sum_{\mu\in\Lambda}c_{\mu}e_{\mu}\in \mathcal{D}'_{\mathfrak{p}_{\Lambda^{*}},(\omega)}$. In analogy to (\ref{eqsFL}), we write
\begin{equation}
\label{eqsFl}
|g|_{\mathcal{F} l^{q,\Gamma}_{v,\Lambda}} :=\left\|(c_\mu)_{\mu\in\Gamma\cap \Lambda} \right\|_{l^{q}_{v}(\Gamma\cap\Lambda)}=\|(c_\mu1_\Gamma(\mu))_{\mu\in\Lambda} \|_{l^q_{v}(\Lambda)}.
\end{equation}
Using (\ref{eqsFl}), the condition $(i)$ from Theorem \ref{main} might be restated as
$$|(\varphi f)_{\mathfrak{p}_{\Lambda^*}}|_{\mathcal{F} l^{q,\Gamma}_{v,\Lambda}}<\infty,$$
and if this is the case we shall say that $f$ is $\mathcal{F} l^q_{v,\Lambda}$-\emph{microlocally regular} at  $(x_0,\xi_0)$. The wave front set $WF_{\mathcal{F} l^q_{v,\Lambda}}(f)$ can be defined as the complement of the set of  $(x_0,\xi_0)$ such that $f$ is $\mathcal{F} l^q_{v,\Lambda}$-microlocally regular at  $(x_0,\xi_0)$.
With this terminology, we may then rephrase Theorem \ref{main} as the following equality between wave front sets:
\begin{equation}
\label{eqWF1}
WF_{\mathcal{F} L^q_{v}}(f)= WF_{\mathcal{F} l^q_{v,\Lambda}}(f),
\end{equation}
for any lattice $\Lambda$ in $\mathbb{R}^{d}$.

We now discuss several consequences of Theorem \ref{main}. Recall that if $X$ is a linear subspace of $\mathcal{D}_{(\omega)}'(\mathbb{R}^{d})$, its associated local space (on an open subset $\Omega$ of $\mathbb{R}^d$) is $X_{loc}(\Omega)=\{f\in\mathcal{D}_{(\omega)}'(\Omega):\: \varphi f\in X, \: \forall\varphi\in\mathcal{D}_{(\omega)}(\Omega) \}$. Using the $\Lambda$-periodization operator (\ref{eqperiodization}), one may talk about local spaces with respect to vector spaces of $\Lambda$-periodic ultradistributions. Indeed, if $Y$ is a linear subspace of  $\mathcal{D}'_{\mathfrak{p}_{\Lambda},(\omega)}$, we set $Y_{loc}(\Omega)=\{f\in\mathcal{D}_{(\omega)}'(\Omega):\: (\varphi f)_{\mathfrak{p}_{\Lambda}}\in Y, \: \forall \varphi\in\mathcal{D}_{(\omega)}(\Omega) \}$. If $X$ and $Y$ are t.v.s. of ultradistributions, the topologies of $X_{loc}(\Omega)$ and $Y_{loc}(\Omega)$ can be defined in the canonical way. Employing a standard partition of the unity argument, Theorem \ref{main} immediately yields:

\begin{corollary}
\label{clocspaces} Let $v\in\mathscr{M}_{(\omega)}$. Then, $(\mathcal{F} L^q_{v})_{loc}(\Omega)=(\mathcal{F} l^q_{v,\Lambda})_{loc}(\Omega)$ topologically, for any  lattice $\Lambda$ in $\mathbb{R}^d$.
\end{corollary}

It should be pointed out that Corollary \ref{clocspaces} remains valid even if we remove the assumption $(\alpha_0)$ on $\omega$.

The equality (\ref{eqWF1}) can be generalized to wave front sets of sup- and inf-types \cite{j-p-t-t,Pilipovictwo}. Indeed, let $(v_j)=(v_j)_{j\in J}$ and $(q_j)=(q_j)_{j\in J}$ be two indexed families with $v_j\in\mathscr{M}_{(\omega)}$ and $q_j\in[1,\infty]$, $\forall j\in J$. The wave front sets of $f\in\mathcal{D}_{(\omega)}'(\Omega)$ of inf-type with respect to the families of Fourier-Lebesgue spaces $(\mathcal{F} L^{q_j}_{v_j})_{j\in J}$ and $(\mathcal{F} l^{q_j}_{v_j,\Lambda})_{j\in J}$ are given by
$$
WF^{\text{inf}}_{(\mathcal{F} L^{q_j}_{v_j})}(f)=\bigcap_{j\in J}WF_{\mathcal{F} L^{q_j}_{v_j}}(f) \quad \mbox{and} \quad WF^{\text{inf}}_{(\mathcal{F} l^{q_j}_{v_j,\Lambda})}(f)=\bigcap_{j\in J}WF_{\mathcal{F} l^{q_j}_{v_j,\Lambda}}(f).
$$
On the other hand, the wave front set $WF^{\text{sup}}_{(\mathcal{F} L^{q_j}_{v_j})}(f)$ is defined as the complement in $\Omega\times(\mathbb{R}^d\setminus\{0\})$ of the set of points $(x_0,\xi_0)$ such that there are an open conic neighborhood $\Gamma$ of $\xi_0$ and $\varphi\in\mathcal{D}_{(\omega)}(\Omega)$ with $\varphi(x_0)\neq 0$ (both independent of the index $j$) such that $|\varphi f|_{\mathcal{F} L^{q_j,\Gamma}_{v_j}}<\infty$, for each $j\in J$.
A similar definition (employing the seminorms (\ref{eqsFl})) applies to the wave front of discrete type $WF^{\text{sup}}_{(\mathcal{F} l^{q_j}_{v_j,\Lambda})}(f)$; in this case one should also assume that $\varphi\in \mathcal{D}_{(\omega)}(x_0+U)$ where $U$ is as in Theorem \ref{main}. We obtain that following stronger version of (\ref{eqWF1}) as a corollary of the proofs of Theorem \ref{main} and Lemma \ref{mainlemma}.

\begin{theorem}\label{mainsup}
Let $v_j \in \mathscr{M}_{(\omega)}$ and $q_j \in [1, \infty]$, where $j$ runs over an index set $J$. For $f\in\mathcal{D}_{(\omega)}'(\Omega)$, we have
\begin{equation}
\label{eqWF2}
WF^{\textnormal{inf}}_{(\mathcal{F} L^{q_{j}}_{v_j})}(f)= WF^{\textnormal{inf}}_{(\mathcal{F} l^{q_j}_{v_j,\Lambda})}(f) \quad \mbox{and}\quad WF^{\textnormal{sup}}_{(\mathcal{F} L^{q_{j}}_{v_j})}(f)= WF^{\textnormal{sup}}_{(\mathcal{F} l^{q_j}_{v_j,\Lambda})}(f),
\end{equation}
for any lattice $\Lambda$ in $\mathbb{R}^{d}$.
\end{theorem}

We end this section we a discrete characterization of    $WF_{\{\omega\}}$ and $WF_{(\omega)}$. If $f\in \mathcal{D}_{(\omega)}'(\Omega)$, these wave front sets can be defined as
$$
WF_{\{\omega\}}(f)=WF^{\textnormal{inf}}_{(\mathcal{F} L^{\infty}_{v_\lambda})}(f)  \quad \mbox{and} \quad WF_{(\omega)}(f)=WF^{\textnormal{sup}}_{(\mathcal{F} L^{\infty}_{v_\lambda})}(f),
$$
with the weights $v_{\lambda}(\xi)=e^{\lambda\omega(\xi)}$, $\lambda>0$. If  $ (x_0, \xi_0)\notin WF_{\{\omega\}}(f)$ (resp. $ (x_0, \xi_0)\notin WF_{(\omega)}(f)$), one says that $f$ is $\{\omega\}$-microlocally regular at  $(x_0, \xi_0)$ (resp. $(\omega)$-microlocally regular). We point out that when $f\in\mathcal{D}'_{\{\omega\}}(\Omega)$, the ultradistribution is $\{\omega\}$-microlocally regular ($(\omega)$-microlocally regular) at $(x_0,\xi_0)$ if and only if there are an conic open neighborhood $\Gamma$ of $\xi_0$ and $\varphi\in \mathcal{D}_{\{\omega\}}(\Omega)$ with $\varphi(x_0)\neq 0$ such that for some $\lambda > 0$ (for every $\lambda > 0$)
\begin{equation}
\label{eq4.5}  \sup_{\xi \in \Gamma} |\widehat{\varphi f}(\xi)| e^{\lambda \omega(\xi)} < \infty,
\end{equation}
so that our definition agrees with the one used in \cite{f-g-j,Pilipovictwo,Rodino}. Summing up, we obtain the ensuing result, a corollary of (\ref{eqWF2}).

\begin{corollary}
\label{corollarywfomega}
Let $f \in \mathcal{D}_{(\omega)}'(\Omega)$ and let $\Lambda$ be a lattice in $\mathbb{R}^{d}$. Suppose that $U$ is an  open convex neighborhood of the origin such that $U\cap\Lambda^{*}=\{0\}$ and $x_0+U\subseteq\Omega$. Then, $f$ is $\{\omega\}$-microlocally regular $($$(\omega)$-microlocally regular$)$ at the point $(x_0,\xi_0)\in\Omega\times ( \mathbb{R}^d \setminus \{ 0 \})$ if and only if
there are an open conic neighborhood $\Gamma$ of $\xi_0$ and $\varphi \in \mathcal{D}_{(\omega)}(x_0+U)$ with $\varphi(x_0)\neq0$ such that for some $\lambda > 0$ $($for every $\lambda >0$$)$
\begin{equation}
\label{eq4.6}
\sup_{\mu \in \Gamma\cap \Lambda} |\widehat{\varphi f}(\mu)| e^{\lambda \omega(\mu)} < \infty.
\end{equation}
Moreover, if $f\in\mathcal{D}'_{\{\omega\}}(\Omega)$ the regularity assumption on the $\varphi$  witnessing $(\ref{eq4.6})$  may be relaxed to $\varphi \in \mathcal{D}_{\{\omega\}}(x_0+U)$.
\end{corollary}

Note that if a weight sequence $(M_p)_{p}$ satisfies $(M.1)$, $(M.2)$, $(M.3)'$, and $(M.4)$, then Theorem \ref{th WF non-quasianalytic} from the introduction turns out to be a particular case of Corollary \ref{corollarywfomega}; we shall however derive the general version of Theorem \ref{th WF non-quasianalytic} in Section \ref{section quasianalytic}, where the quasianalytic case will also be treated. Nonetheless, it is should be mentioned that Corollary \ref{corollarywfomega} already covers important cases such as  $\omega(\xi) = |\xi|^{1/s}$, $s > 1$, which corresponds to $M_p=(p!)^{s}$; thus it provides a discrete characterization of $\{s\}$- and $(s)$-microregularity \cite{Rodino}. Furthermore, Corollary \ref{corollarywfomega} also includes results by Ruzhansky and Turunen on the toroidal $C^{\infty}$-wave front set \cite[Thm. 7.4]{RT}, while Theorem \ref{mainsup} covers all results from \cite{d-m-s,Maksi}.

\section{Quasianalytic wave front sets}
\label{section quasianalytic}
This last section is devoted to wave front sets defined via weight sequences.

We consider two sequences $(M_p)_{p \in \mathbb{N}}$ and $(N_p)_{p \in \mathbb{N}}$ of positive real numbers (with $M_0 = N_0 = 1$). The sequence $(M_p)_p$ is assumed to satisfy $(M.1)$, $(M.2)'$, $(M.3)'$, while $(N_p)_p$ satisfies $(M.1)$, $(M.2)'$, and in addition
\begin{itemize}
\item [$(M.5)$]
$p!\subset N_{p}$ in the Roumieu case, \quad or \quad $ p!\prec N_p$ in the Beurling case.
\end{itemize}
The sequence $(N_p)_p$ may be quasianalytic, namely, $\sum_{p=1}^{\infty} N_{p-1}/{N_{p}}=\infty$, and will be used to measure the micro\-regularity of an $M_p$-ultradistribution. The functions $M$ and $N$ stand for the associated functions of $(M_p)_p$ and $(N_p)_p$, respectively.

We shall follow H\"{o}rmander's approach to quasianalytic wave front sets \cite[Sect. 8.4]{Hormander}, but we slightly modify it to include wave fronts of $M_p$-ultradistributions. Let $f \in {\mathcal{D}^{(M_p)}}'(\Omega)$. We begin with the Roumieu case. The ultradistribution $f$ is said to be $\{N_p\}$-\emph{microlocally regular} at the point $(x_0,\xi_0)\in \Omega\times (\mathbb{R}^d \setminus \{ 0 \})$ if there are an open conic neighborhood $\Gamma$ of $\xi_0$, an open neighborhood $V$ of $x_0$ and a bounded sequence $(f_p)_{p \in \mathbb{N}}$  in ${\mathcal{E}^{(M_p)}}'(\Omega)$, with $f_p= f$ on $V$ for all $p \in \mathbb{N}$, such that for some $A,C > 0$
\begin{equation}
\label{eqWFseq}
  \sup_{\xi \in \Gamma} |\widehat{f_p} (\xi)| |\xi|^p  < AC^{p}N_p, \qquad \forall p \in \mathbb{N}.
 \end{equation}
The wave front set $WF_{\{N_p\}}(f)$ then consists of all those $(x_0,\xi_0)\in \Omega\times (\mathbb{R}^d \setminus \{ 0 \})$ such that $f$ is \emph{not} $\{N_p\}$-microlocally regular at $(x_0,\xi_0)$. For classical Schwartz distributions $f \in {\mathcal{D}}'(\Omega)$, $WF_{\{N_p\}}(f)$ agrees with H\"{o}rmander's notion $WF_L(f)$, where $L_p = N_p^{1/p}$ (and our assumptions on $N_p$ turn out to be the same as those considered in \cite[Sect. 8.4]{Hormander} for $L_p$). In particular, for $N_p= p!$, we obtain the analytic wave front set $WF_A(f):=WF_{\{p!\}}(f)$.

 One defines $(N_p)$-\emph{microregularity} in a similar fashion (namely, by asking that (\ref{eqWFseq}) holds for all $C>0$ and some $A=A_C>0$). The definition of the Beurling wave front set  $WF_{(N_p)}(f)$ should be clear.

We need the notion of an analytic cut-off sequence \cite{Hormander,Petschze2} in order to move further. A sequence
$(\chi_p)_{p\in \mathbb{N}}$ in $\mathcal{D}^{(M_p)}(\Omega)$ is called an $(M_p)$-analytic cut-off sequence supported in $\Omega$ if
\begin{itemize}
\item[$(a)$] $(\chi_p)_p$ is a bounded sequence in $\mathcal{D}^{(M_p)}(\Omega)$,
\item[$(b)$] $(\exists C > 0)(\forall h > 0)(\exists A_h > 0)$
\[ \|\chi_p^{(\alpha)}\|_{\mathcal{E}^{\{M_p\},h}(L)} \leq A_h(Cp)^{|\alpha|}, \qquad \forall p \in \mathbb{N},\ |\alpha| \leq p,\]
\end{itemize}
where $L \Subset \Omega$ is such that $\operatorname{supp} \chi_p \subseteq L$ for all $p \in \mathbb{N}$. We call $(\chi_p)_p$ an analytic cut-off sequence for $K\Subset\Omega$ if
\begin{itemize}
\item[$(c)$] there exists an open neighborhood $V$ of $K$ such that $\chi_p \equiv 1$ on $V$ for all $p \in \mathbb{N}$.
\end{itemize}
If $K = \{x_0\}$ is a singleton set, we shall simply say that $(\chi_p)_p$  is an analytic cut-off sequence for $x_0$. Likewise, one may also define an $\{M_p\}$-analytic cut-off as a bounded sequence $(\chi_p)_{p\in \mathbb{N}}$ in $\mathcal{D}^{\{M_p\}}(\Omega)$ such that the property $(b)$ is asked to hold just for some $h,A_h>0$.

\begin{lemma}
Let $K \Subset \mathbb{R}^d$. For every open neighborhood $W$ of $K$ there exists an $(M_p)$-analytic cut-off sequence for $K$ supported in $W$.
\end{lemma}
\begin{proof}
 Set $d(K, \partial W) = 4\varepsilon > 0$. By \cite[Thm. 1.4.2]{Hormander} there is a sequence $(\chi_p)_p$ in $\mathcal{D}(\mathbb{R}^d)$ and $C>0$ such that
 \[ \|\chi_p^{(\alpha)}\|_{L^\infty} \leq (Cp)^{|\alpha|}, \qquad \forall p \in \mathbb{N}, |\alpha| \leq p,\]
 $\chi_p \equiv 1$ on $K+ \bar{B}(0,2\varepsilon)$ and $\operatorname{supp} \chi_p \subseteq  K+ \bar{B}(0,3\varepsilon)$ for all $p \in \mathbb{N}$. Let $\varphi \in \mathcal{D}^{(M_p)}(\mathbb{R}^d)$ with $\operatorname{supp} \varphi \subseteq B(0, \varepsilon)$ and $\int_{\mathbb{R}^d}\varphi(x) \mathrm{d}x = 1$. It is clear that $(\chi_p \ast \varphi)_p$  satisfies all requirements.
\end{proof}

 We can now state the main result of this section.
\begin{theorem}\label{mainqa}
 Let $\Lambda$ be a lattice in $\mathbb{R}^{d}$, $f \in {\mathcal{D}^{(M_p)}}'(\Omega)$, and $(x_0, \xi_0) \in \Omega \times (\mathbb{R}^d \backslash \{ 0 \})$. Suppose that $U$ is an  open convex neighborhood of the origin such that $U\cap\Lambda^{*}=\{0\}$ and $x_0+U\subset \Omega$. Then, the following statements are equivalent:\begin{itemize}
\item[$(i)$]  There are an open conic neighborhood $\Gamma$ of $\xi_0$, an open neighborhood $V \subseteq x_0+U$ of $x_0$, and a bounded sequence $(f_p)_{p \in \mathbb{N}}$  in ${\mathcal{E}^{(M_p)}}'( x_0 + U)$, with $f_p= f$ on $V$ for all $p \in \mathbb{N}$, such that for some $A,C > 0$ $($for every $C>0$ there is $A>0$$)$

\[  \sup_{\mu \in \Gamma \cap \Lambda} |\widehat{f_p} (\mu)| |\mu|^p  \leq AC^{p}N_p, \qquad \forall p \in \mathbb{N}.  \]
\item[$(ii)$] There are an open conic neighborhood $\Gamma$ of $\xi_0$ and an $(M_p)$-analytic cut-off sequence $(\chi_p)_p$ for $x_0$  supported in $x_0 + U$ such that for some $A,C > 0$ $($for every $C>0$ there is $A>0$$)$
\[  \sup_{\mu \in \Gamma \cap \Lambda} |\widehat{\chi_p f} (\mu)| |\mu|^p  \leq AC^{p}N_p, \qquad \forall p \in \mathbb{N}.  \]
\item[$(iii)$] There are an open conic neighborhood $\Gamma$ of $\xi_0$ and an $(M_p)$-analytic cut-off sequence $(\chi_p)_p$  for $x_0$ such that for some $A,C > 0$ $($for every $C>0$ there is $A>0$$)$
\[  \sup_{\xi \in \Gamma} |\widehat{\chi_p f} ( \xi)| |\xi|^p  \leq AC^{p}N_p, \qquad \forall p \in \mathbb{N}.  \]
\item[$(iv)$] $f$ is $\{N_p\}$-microlocally regular $($$(N_p)$-microlocally regular$)$ at $(x_0,\xi_0)$.
\end{itemize}
\smallskip
\smallskip

\noindent If additionally $(N_p)_p$ satisfies $(M.3)'$,
then these statements are also equivalent to any of the following two conditions:
\begin{itemize}
\item [$(v)$] There are an open conic neighborhood $\Gamma$ of $\xi_0$ and $\varphi \in \mathcal{D}^{(M_p)}(x_0+U)$ with $\varphi\equiv1$ in a neighborhood of $x_0$  such that for some $r > 0$ $($for every $r >0$$)$
$$
\sup_{\mu \in \Gamma\cap \Lambda} |\widehat{\varphi f}(\mu)| e^{N(r\mu)} < \infty.
$$
\item [$(vi)$] There are an open conic neighborhood $\Gamma$ of $\xi_0$ and $\varphi \in \mathcal{D}^{(M_p)}(x_0+U)$ with $\varphi\equiv1$ in a neighborhood of $x_0$  such that for some $r > 0$ $($for every $r >0$$)$
$$
\sup_{\xi \in \Gamma} |\widehat{\varphi f}(\xi)| e^{N(r\xi)} < \infty.
$$
\end{itemize}
\end{theorem}

Observe that both Theorem \ref{th WF non-quasianalytic} and Theorem \ref{th WF analytic} are contained in Theorem \ref{mainqa}. The rest of this section is dedicated to give a proof of Theorem \ref{mainqa}. We divide its proof into several intermediate lemmas.
\begin{lemma}\label{bounded}
Let $(\chi_p)_p$ be an $(M_p)$-analytic cut-off sequence and $B$ a bounded subset of $\mathcal{E}^{(p!)}(\mathbb{R}^d)$. Then,
\begin{itemize}
\item[\mbox{}] $(\exists C > 0)(\forall h > 0)(\exists C_h > 0)$
\[ \sup_{\psi \in B}\sup_{\xi \in \mathbb{R}^d} |\widehat{ \chi_p\psi}(\xi)| |\xi|^{p} e^{M(|\xi|/h)} \leq C_h (Cp)^p, \qquad \forall  p \in \mathbb{N}.\]
\end{itemize}
\end{lemma}
\begin{proof}
Let $K \Subset \mathbb{R}^d$ be such that $\operatorname{supp} \chi_p \subseteq K$ for all $p \in \mathbb{N}$. A straightforward computation yields
\begin{itemize}
\item[\mbox{}] $(\exists C > 0)(\forall h > 0)(\exists C_h > 0)$
\[ \sup_{\psi \in B} \|(\chi_p\psi)^{(\alpha)}\|_{\mathcal{E}^{(M_p),h}(K)} \leq C_h (Cp)^{|\alpha|}, \qquad \forall p \in \mathbb{N}, |\alpha| \leq p.\]
\end{itemize}
Hence for all $p, q \in \mathbb{N}$, $\psi \in B$ and $h > 0$ we have that
\begin{align*}
|\widehat{ \chi_p\psi}(\xi)| |\xi|^{p+q} &\leq d^{\frac{p+q}{2}} |K| \sup_{|\alpha| = p} \sup_{|\beta|=q} \sup_{x \in K}|(\chi_p\psi)^{(\alpha+\beta)}(x)| \\
 &\leq |K| C_{h/\sqrt{d}}(\sqrt{d}Cp)^p h^qM_q,
 \end{align*}
 where $|K|$ is the Lebesgue measure of $K$. Consequently,
 \[ |\widehat{ \chi_p\psi}(\xi)| |\xi|^{p} \leq |K| C_{h/\sqrt{d}}(\sqrt{d}Cp)^p \inf_{q \in \mathbb{N}} \frac{h^qM_q}{|\xi|^q} = |K|  C_{h/\sqrt{d}}(\sqrt{d}Cp)^pe^{-M(|\xi|/h)}.\]
\end{proof}
\begin{lemma}\label{lemmaqa}
Condition $(ii)$ from Theorem \ref{mainqa} implies that there are an open conic neighborhood $\Gamma_1$ of $\xi_0$, an open neighborhood $U_1 \subseteq U$ of the origin, and an $(M_p)$-analytic cut-off sequence $(\kappa_p)_p$ for $x_0$ supported in $x_0 + U_1$ such that for every bounded set $B$ in $\mathcal{E}^{(p!)}(\mathbb{R}^d)$ there are $A,C > 0$ $($for every $C>0$ there is $A>0$$)$
for which
\[  \sup_{\psi \in B}\sup_{\mu \in \Gamma_1 \cap \Lambda}  |\widehat{\kappa_p \psi f} (\mu)| |\mu|^p  \leq A C^{p}N_p, \qquad \forall p \in \mathbb{N}.  \]
\end{lemma}
\begin{proof}
We choose an open conic neighborhood $\Gamma_1$ of $\xi_0$ such that $\overline{\Gamma}_1 \subseteq \Gamma \cup \{ 0 \}$. Let  $0 < c < 1$ be smaller than the distance between $\partial \Gamma$ and the intersection of $\Gamma_1$ with the unit sphere. Hence $\{y \in \mathbb{R}^d \,:\, (\exists \xi \in \Gamma_1)(|\xi -y|\leq c|\xi|) \} \subseteq \Gamma$. Next, choose an open neighborhood $U_1 \subseteq U$ of the origin such that $\chi_p \equiv  1$ on  $x_0 + U_1$ for all $p \in \mathbb{N}$. Let  $(\kappa_p)_p$  be an analytic cut-off sequence for $x_0$ supported in $x_0 + U_1$. Let $B$ be a bounded set in $\mathcal{E}^{(p!)}(\mathbb{R}^d)$. Since $x_0+U$ is contained in some fundamental region $I_{\Lambda^{\ast}}$ of the dual lattice $\Lambda^*$,we have that for an arbitrary $g \in \mathcal{E}^{(M_p)'}(\mathbb{R}^{d})$ having support in  the interior of $I_{\Lambda^{\ast}}$ the Fourier coefficients of its $\Lambda^*$-periodization are given by $(\widehat{g}(\mu))_{\mu\in\Lambda}$. In particular, using the fact that $(\kappa_
 p \psi f
 )_{\mathfrak{p}_{\Lambda^*}}= (\kappa_p\psi)_{\mathfrak{p}_{\Lambda^*}}(\chi_p f)_{\mathfrak{p}_{\Lambda^*}}$ for all $p \in \mathbb{N}$ and $\psi \in B$, we obtain that
\[
 |\widehat{\kappa_p \psi f} (\mu)| = \left| \sum_{\beta \in \Lambda} \widehat{\chi_pf}(\mu - \beta) \widehat{\kappa_p\psi}(\beta)\right| \leq I_{1,p}(\mu) + I_{2,p}(\mu),
\]
where
\[
I_{1,p}(\mu) = \underset{\beta \in \Lambda}{\sum_{|\beta| \leq c|\mu|}} |\widehat{\chi_pf}(\mu-\beta)| |\widehat{\kappa_p\psi}(\beta)|
\]
and
\[
I_{2,p}(\mu) = \underset{\beta \in \Lambda}{\sum_{|\beta| > c|\mu|}} |\widehat{\chi_pf}(\mu-\beta)| |\widehat{\kappa_p\psi}(\beta)|
.
\]
Due to the fact that the set $\{\kappa_p\psi \,: \, p \in \mathbb{N}, \psi \in B\}$ is bounded in $\mathcal{D}^{(M_p)}(\Omega)$, there exists $D > 0$, independent of $p$ and $\psi$, such that
\[ I_{1,p}(\mu) = \underset{\beta \in \Lambda}{\sum_{|\mu-\beta| \leq c|\mu|}} |\widehat{\chi_pf}(\beta)| |\widehat{\kappa_p\psi}(\mu-\beta)| \leq D \underset{\beta \in \Lambda}{\sup_{|\mu-\beta| \leq c|\mu|}} |\widehat{\chi_pf}(\beta)|
 \]
 Since $|\mu-\beta| \leq c|\mu|$ implies $|\beta| \geq (1-c)|\mu|$, we have that
 \begin{align*}
  \sup_{\psi \in B}\sup_{\mu \in \Gamma_1 \cap \Lambda} I_{1,p}(\mu) |\mu|^p &\leq D \sup_{\mu \in \Gamma_1 \cap \Lambda} |\mu|^p \sup_{\beta \in \Lambda}\sup_{|\mu-\beta| \leq c|\mu|} |\widehat{\chi_pf}(\beta)|\\
  & \leq \frac{D}{(1-c)^p} \sup_{\beta \in \Gamma \cap \Lambda}|\beta|^p|\widehat{\chi_pf}(\beta)| \\
  & \leq D A \left(\frac{C}{1-c} \right)^p N_p.
 \end{align*}
We now estimate $I_{2,p}(\mu)$.  The boundedness of $(\chi_p)_p$ implies that there are $D', h > 0$ such that for all $p  \in \mathbb{N}$
 \[  |\widehat{\chi_pf}(\xi)| \leq D' e^{M(|\xi|/h)}, \qquad \forall \xi \in \mathbb{R}^d.\]
Since $|\mu-\beta| \leq (1+ c^{-1})|\beta|$ for $|\beta| \geq c |\mu|$, we have that
\begin{align*}
\sup_{\psi \in B}\sup_{\mu \in \Gamma_1 \cap \Lambda} I_{2,p}(\mu) |\mu|^p &\leq D'  \sup_{\psi \in B}\sup_{\mu \in \Gamma_1 \cap \Lambda} |\mu|^p \underset{\beta \in \Lambda}{\sum_{|\beta| \geq c|\mu|} }e^{M(|\mu-\beta|/h)}|\widehat{\kappa_p\psi}(\beta)|
\\
&
\leq   \frac{D'}{c^p} \sup_{\psi \in B}\sum_{\beta \in \Lambda} e^{M((1+ c^{-1})|\beta|/h)} |\beta|^p|\widehat{\kappa_p\psi}(\beta)|.
 \end{align*}
 The result now follows from Lemma \ref{bounded}, \cite[Prop. 3.4]{Komatsu}, and the assumption $(M.5)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{mainqa}.] The implications $(iii) \Rightarrow (iv)$ and
$(iv) \Rightarrow (i)$, $(v)\Rightarrow(i)$, and $(vi) \Rightarrow(v)$ are trivial. The proof of
$(i) \Rightarrow (ii)$ is similar to the first part of that of \cite[Lemma 8.4.4]{Hormander}, so we omit it. Let us now show
$(ii) \Rightarrow (iii)$. For it,  let $I_\Lambda$ be a fundamental region for $\Lambda$ with $0 \in I_\Lambda
$ and let $\Gamma_1$ and $(\kappa_p)_p$ be as in Lemma \ref{lemmaqa}. Choose an open conic neighborhood $\Gamma_2$ of $\xi_0$ such that $\overline{\Gamma}_2 \subseteq \Gamma_1 \cup \{ 0 \}$.  Fix $r > 0$ such
that $\Gamma_2 \cap \{ \xi \in \mathbb{R}^d \, | \, |\xi| \geq r \} \subseteq (\Gamma_1 \cap \Lambda) + I_\Lambda$. Since $I_{\Lambda}$ is bounded, there is $D > 0$ such that $|\mu+t|^p \leq D^p |\mu|^p$ for all $p \in \mathbb{N}$, $t \in I_\Lambda$ and $\mu \in \Lambda\setminus\{0\}$.
Hence Lemma \ref{lemmaqa} and the fact that $B = \left \{ e_{-t} \, | \, t \in I_\Lambda \right\}$ is a bounded subset of $\mathcal{E}^{(p!)}(\mathbb{R}^d)$ imply that
\begin{align*}
\sup_{\xi \in \Gamma_2, |\xi| \geq r } |\widehat{\kappa_p f}(\xi)||\xi|^p & \leq \sup_{\mu \in \Gamma_1 \cap \Lambda} \sup_{t \in I_\Lambda}  |\widehat{\kappa_p f}(\mu+t)||\mu+t|^p \\
&\leq D^p\sup_{t \in I_\Lambda}\sup_{\mu \in \Gamma_1 \cap \Lambda}|\mathcal{F}(\kappa_p e_{-t} f)(\mu)||\mu|^p \leq A(DC)^pN_p.
\end{align*}

It remains to establish $(iv) \Rightarrow(vi)$ under the additional assumption that $(N_p)_p$ satisfies $(M.3)'$. So assume $(iv)$ and let $\Gamma_1$ and $c$ be as in the proof of Lemma  \ref{lemmaqa}. We use the auxiliary weight sequence $Q_p=\min_{0\leq q\leq p}M_q N_{p-q}$, with associated function $Q$. A result by Roumieu (\cite[Lemm. 3.5]{Komatsu}) ensures that $(Q_p)_p$ satisfies $(M.1)$ and that $Q=M+N$; thus \cite[Prop. 3.4]{Komatsu} implies that $(Q_p)_p$ also fulfills $(M.2)'$. Select any $V_1$ with $\overline{V}_1\Subset V$ and $\varphi\in \mathcal{D}^{(Q_p)}(V)$ with $\varphi\equiv1$ on $V_1$. Furthermore, using the boundedness of $(f_p)_{p}$, find constants $D, h>0$ such that
 \[ \|\widehat{\varphi}\|_{L^{1}}\leq D \quad \mbox{and}
 \quad \sup_{\xi\in\mathbb{R}^{d},\: p\in\mathbb{N} } e^{-M(|\xi|/h)}|\widehat{f_p}(\xi)| \leq D.
 \]
Set $R=\max{\{ h^{-1}(1+ c^{-1}),(cC)^{-1}(1-c)\}}.$ Reasoning as in the proof of Lemma \ref{lemmaqa}, we obtain, for all $p\in\mathbb{N}$ and $\xi\in \Gamma_1$,
\begin{align*}
\left| \xi\right|^{p}|\widehat{\varphi f}(\xi)|&
=\left| \xi\right|^{p}|\widehat{\varphi f_p}(\xi)|
\\
&\leq
DA \left(\frac{C}{1-c}\right)^{p} N_p+DN_p\int_{|\eta| > c |\xi| }e^{M(h^{-1}|\xi - \eta|)}\frac{|\eta|^p}{c^{p}N_p} |\widehat{\varphi}(\eta)|\mathrm{d}\eta
\\
&\leq
D\left(\frac{C}{1-c}\right)^{p} N_p\left(A +\int_{\mathbb{R}^d}e^{M(h^{-1}(1+ c^{-1})|\eta|)+N((cC)^{-1}(1-c)|\eta|)} |\widehat{\varphi}(\eta)| \mathrm{d}\eta\right)
\\
&\leq
D\left(\frac{C}{1-c}\right)^{p} N_p\left(A +\int_{\mathbb{R}^d}e^{Q(R|\eta|)} |\widehat{\varphi}(\eta)|\mathrm{d}\eta \right).
\end{align*}
The last integral is finite because of \cite[Prop. 3.4]{Komatsu} and so $\displaystyle\sup_{\xi \in \Gamma_1} |\widehat{\varphi f}(\xi)| e^{N(r\xi)} < \infty$, with $r=(1-c)/C$.
\end{proof}

We conclude the article with some remarks.
\begin{remark}\label{rk1Fs} Let $f\in {\mathcal{D}^{\{M_p\}}}'(\Omega)$ ($f\in \mathcal{D}'(\Omega)$). Then, the requirements on the sequence $(\chi_p)_p$ and the test function $\varphi$ from Theorem \ref{mainqa} can be relaxed to:   $(\chi_p)_p$ is an $\{M_p\}$-analytic (analytic) cut-off sequence  for $x_0$ and $\varphi \in \mathcal{D}^{\{M_p\}}(x_0+U)$ ($\varphi \in \mathcal{D}(x_0+U)$). Furthermore, the properties of $(f_p)_p$ in condition $(i)$ of Theorem \ref{mainqa} and in the definition of $\{N_p\}$- and $(N_p)$-microregularity  can be strengthened to: $(f_p)_p$ is bounded in ${\mathcal{E}^{\{M_p\}}}'(x_0+U)$ (in $\mathcal{E}'(x_0+U)$) and ${\mathcal{E}^{\{M_p\}}}'(\Omega)$ ($\mathcal{E}'(\Omega)$), respectively. The proofs of these assertions are straightforward modifications of the arguments given in this section and are therefore left to the reader.
\end{remark}

\begin{remark}
If the sequence  $(M_p)_p$ additionally satisfies $(M.4)''$, then one might simply ask $\varphi(x_0)\neq0$ for $\varphi \in \mathcal{D}^{(M_p)}(x_0+U)$ ($\varphi \in \mathcal{D}^{\{M_p\}}(x_0+U)$ if $f\in {\mathcal{D}^{\{M_p\}}}'(\Omega)$)  occurring in $(v)$ from Theorem \ref{mainqa}. In fact, all this is a just consequence of the fact that $\mathcal{E}^{\{M_p\}}(\Omega)$ and $\mathcal{E}^{(M_p)}(\Omega)$ become inverse closed under $(M.4)''$.  In particular, that is the case when $(M_p)_p$ fulfills $(M.3)$, as already pointed out in Subsection \ref{subsection sequences}.
\end{remark}

redpajama_set_name: RedPajamaArXiv


Dr. Sprain's research program focuses on democratic engagement, studying how specific communication practices facilitate and inhibit democratic action. Because outreach and praxis are crucial to democratic engagement, much of her research is collaborative, funded, and focused on the theory–practice interface. Informed by work as a deliberative practitioner with the Colorado State University (CSU) Center for Public Deliberation (CPD), her research bridges conventional divisions among normative theory, empirical research, and practice. By working at the theory–practice interface, Dr. Sprain's research informs and improves democratic engagement on pressing social issues, such as climate change, water conflict, and community resilience.


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Army Chief Bipin Rawat claims demography of entire Pok changed by Pak
in Defense, Featured, Latest, National, Trending
Army Chief Bipin Rawat says Indian army is capable of using strike drones in Jammu and Kashmir if the country will accept the consequences of such actions (Image Courtesy-DNA India)
Indian Army Chief General Bipin Rawat stated that Pakistan has very cleverly changed the complete demography of Pakistan-occupied Kashmir (PoK) including Gilgit-Baltistan, while delivering the Yashwantrao Chavan Memorial Lecture in Delhi on Wednesday.
The Army Chief General also emphasized in his speech that the identity of Kashmiris on the other side of the Line of Control has systematically been eroded by Islamabad.
On Wednesday, while he was delivering the lectures, he asserted, "Every time something happens on our side, of Line of Control, we must always address it to say that it is also going to have a radical effect on other side (PoK). The issue is, on the other side, the complete demography has changed."
"Pakistan has very cleverly changed the complete demography of so-called Pakistan-occupied Kashmir, Gilgit-Baltistan. So, one is not very sure as to who is an actual Kashmiri, whom we are addressing these issues to," added the Army Chief General on the sidelines of that occasion.
Further he said, "People from Gilgit-Baltistan (part of PoK) are also now being taken over gradually by Islamabad. So, to say that there is an identity between our side (of LoC) of Kashmir and the other side, then the identity thing has gradually been eroded very cleverly by the Pakistanis. That is an issue we have to look at."
The army chief also claimed that the tolerance levels of the people in Jammu and Kashmir have 'reduced and perceived alienation' as the youth in the region are being 'indoctrinated to act irrationally.'
Bipin Rawat also highlighted that "the proliferation of social media has further compounded the challenge and contributes towards the spread of an anti-establishment narrative."
Further he said that a perception in Jammu and Kashmir is based on whose story you are listening to. Thus, making peace narrative is very different. These narratives are built around the constructs of radicalization, alienation, agitation, religious fundamentalism and anti-nationalism on one hand and the perception of a Kashmiri common man on the other.
In between his lectures, talking about the state J&K, Bipin Rawat explained that things will be brought under control and are being brought under control, but sustained pressure needs to be maintained.
The Army Chief General also expressed concern over allowing funeral processions of terrorists, saying that those who are glorifying terrorists as brave hearts are 'possibly trying encouraging more people to join the terrorist ranks.'
The army chief's comments came after two Central Reserve Police Force or CRPF jawans sustained injuries on Wednesday in a stone-pelting incident in J&K Budgam district. A group of people had resorted to stone pelting during an ongoing encounter between the security forces and the terrorists in Chathergram village, according to reports.
On the same occasion while the Army chief was responding to a question 'if India will follow the footsteps of the United States in using the strike drones to terminate the hostile targets.'
Further he answered that the Indian army is capable of using strike drones in Jammu and Kashmir if the country will accept the consequences of such actions.
He also added that The Indian Army sees no problem in using the strike drones on the potential targets if the people accept their 'mistakes' and 'collateral damages'.
The Army Chief General also highlighted that in India, the way things are moving, 'it's nice to say that we need these drones.'
Further while he was asked on the possibility of the Indian army using the strike drones was about 'across the border.' He said that "Anything we procure (weapons) has got dual use. The first issue is of collateral damage."
And then justifying his remarks, he said that when a country is fighting such kind of war, "You have to accept the collateral damage, it will happen, it is war."
Tags: Drone strike in Gilgit-BaltistanDrone strike in J&KGeneral Bipin Rawat on PoKIndian Army ChiefPakistan Occupied Kashmir
India's first hyper-spectral imaging satellite launched with 30 foreign satellites
Under 'Saubhagya' scheme 15 states achieve 100% electricity: Union Minister RK Singh

redpajama_set_name: RedPajamaCommonCrawl


text:
Actuaries in the Pacific
I started my Pacific tour in Japan. My first appointment, under the big red lantern at the temple in Asakusa, Tokyo, looked like a no-show. While I was waiting, a German businessman spoke to me, I took a photograph for a tourist, and a kimono-clad man smiled at me. As the rain pelted down, I felt like someone in a second-rate detective movie and wondered what the codeword was. The only special package I had brought from England was a bowler hat, acquired at Lock's of St James's, but that was for another contact. There was no actuary in sight.
However, it turned out that there were two other lanterns nearby one of which was much bigger and was at the gate to the entire complex which comprised shrines, shops, houses, restaurants, and a hotel within its grounds. I had been under the wrong lantern.
Friendly people
I had imagined the Japanese to be formal (which they were at first), but in no time they become friendly and easy-going. I had also expected a vast impenetrable city but, once you get the hang of the metro, Tokyo is no more difficult than Paris. A series of sub-cities, each with its own identity, Tokyo is complex, like the people. Fortunately the actuarial market is straightforward. It is a small community and everyone knows everybody else. Of course, that is not to say that the Japanese are gossips. In my experience, they are trustworthy, decent types who do business the way the English did 20 years ago.
For over ten years the Japanese economy was like a sumo wrestler powerful but slow. China has been attracting the bulk of the region's investment while Japan positions itself as the source of technical expertise for Chinese development. The strategy is beginning to bear fruit and growth in Japan is beginning to return, although the restructuring of China's banking system may dampen the newfound confidence.
Many multinational insurers seem to regard Japan as a niche that is not worth the effort in itself, though the recent granting of a reinsurance licence to a big name could change that in future. The insurance market is being gradually deregulated and my assessment is that eventually there will be growth leading to new opportunities for actuaries, perhaps even as soon as next year. Foreign actuarial skills are well regarded, but tackling the language is quite an undertaking.
A niche market
Across the Pacific, yet feeling closer to home culturally, is the niche actuarial market of New Zealand. Historically a country that produces sound actuaries, the profession there is now suffering from legislative change that means saving for retirement is likely to involve property investment. There is still an active actuarial community but the centralisation culture of multinationals often leads to the transfer of systems to Sydney, leaving New Zealand out on a limb. This is a shame, because the economy is going well, the active lifestyle is fun, and the cost of living is very reasonable.
Sydney lived up to its world-class billing in every way for me. The city itself is stunning. Culturally diverse, but with a strong Australian identity, it has positioned itself as the capital of the Pacific. In the actuarial world I believe it is stealing a march on Hong Kong. If China lets its economic treasure tarnish then Sydney will step in. Located in the ideal time zone for the emerging countries, Australia offers a lifestyle to multinational executives that is bound to encourage regional offices to relocate there. Financial institutions have already spread to the leafy suburbs and I would not be surprised to see them go further and develop across the harbour at Manly or beyond, to get the full benefit of the beach. Why not? Work hard, play hard! People jetting around the world are attracted to a base that is beautiful, stylish, and cosmopolitan, but not too congested.
The Australian actuarial community is in flux. Legislative change has damaged the demand for pensions actuaries as it has in New Zealand. The British foray of AMP has engendered caution. Turmoil within the consultancies, some of which seem to have swapped whole teams, has created uncertainty. However, I believe the challenges are being faced up to and opportunities grasped. The sale of regional operations to locals is encouraging entrepreneurial activity. In addition, a new breed of life and non-life insurer is emerging, and the tendency to develop a semi-corporate regional base in Sydney may lead to more opportunities.
The influence of the Chinese is obvious throughout the Pacific, and many tip China to be the economic superpower of the future. Perhaps so, if the growth can be harnessed. However, if democracy takes hold, the ability to develop may be hampered by new political considerations.
04_07_02.pdf


Son 4 apartments for 4 O 5 people. All apartments have a rustic style and offer free Wi-Fi and a private balcony with mountain views. Each room has its own bathroom and TV and air conditioning heating and air conditioning. The living room has wooden beams, sofa bed, TV and DVD player. They also have a private bathroom and a kitchenette well equipped, fridge, microwave and washing machine.
The country-style restaurant serves regional food Mirador. The hotel also has a bar and a garden terrace. In setting wine and olive oil produced, products you can eat on the terrace.
El Mirador Del Soto has a free parking and a tour desk. The A66 motorway is 32 km. Plasencia is a 1 hour drive. La Alberca is 15 km and the Miranda Castañar 11 km.


text:
Oregon ArtsWatch
FULL CIRCLE FUND
Anatomy of a Hug
Chapel Theatre's "Anatomy" lessons
Kat Ramsburg's "Anatomy of a Hug" grapples with the challenge of connection in a fraught mother-daughter relationship.
By: Marty Hughley
Published October 17, 2018, in THEATER
Living with roommates can be tough. Sharing space, overlapping schedules, compromising privacy — it all can be tricky. And if you wind up stuck with someone that, for whatever reason, you're not inclined to like, the situation can get ugly.
Even so, it's a bit of a shock when Amelia, giving a cursory tour of her apartment to an older woman named Sonia, snaps at her that they shouldn't "share any personal information."
But then, you surely could add "mother and daughter" to the list of emotionally loaded living situations.
Chapel Theatre Collective's "Anatomy of a Hug" deals with the difficulty of getting close when you've become a wooden character in your own life story.
And what else might make things especially tense? Let's see…how about if mom's been in prison for killing dad? And the daughter is convinced that she was also a target of the crime? And mom's out now through a compassionate-release program because she has terminal cancer?
That's the potent set-up for Anatomy of a Hug, a well-crafted drama by Los Angeles playwright Kat Ramsburg, on stage now as the debut production from the Chapel Theatre Collective. It's a smart, emotionally perceptive piece of writing, with the acidity of its premise balanced by just enough romantic sweetness.
DramaWatch: Going to the Chapel and we're gonna see theater
Chapel Theatre Collective makes its debut in Milwaukie, the Red Door Project finds a helpful way to "Cop Out," Red Riding Hood goes Shaking the Tree, and more.
Published October 5, 2018, in CULTURE, FEATURED, THEATER
Jason Glick had no intention to run a theater company.
"I'd tried to start a theater company before, in Nashville, when I was in my twenties and I didn't know what not to do," he says. "And I was happy with my place in the community here, with the acting work I was getting at Artists Rep and such. Fundraising, worrying about where the money's coming from for the next show, that kind of stuff was never on my bucket list."
And yet, here he is at the Chapel Theatre in Milwaukie, a performance space that opened earlier this year, preparing the debut show by the resident theater company, the Chapel Theatre Collective.
"It really just fell in my lap," he says, talking before a recent rehearsal of Anatomy of a Hug. "And there was a feeling that we're in the right place at the right time."
Jessica Hillenbrand (clockwise from front right) Murri Lazaroff-Babin, Amanda Vander Hyde and Jason Glick rehearse "Anatomy of a Hug" at Chapel Theatre. Photo: Danielle Weathers.
The place is a refurbished church a reasonable drive from Portland proper, not far off of highways 99 and 224. It has 99 seats on movable risers, a small bar for beer, wine and sodas, a computerized lighting board, and a basement eatery called the Secret Pizza Society (a relative of the Southeast Portland vegan deli Papa G's). The time is an era of increasing competition for too few performance spaces in the city. (The space also is the regular home of the dance company Trip the Dark., and the company Street Scenes also has plays scheduled there later this year.)
The origins of the Chapel Theatre Collective can be traced back to Milagro's 2016 production Davita's Harp, whose cast included, among others, Glick, Danielle Weathers and Illya Torres-Garner. Later, after Torres-Garner, who owns a construction company, had purchased the old chapel near his home, he also happened to be doing work on Glick's house and talked about wanting someone to produce theater in his new space. Glick, a former Theatre Vertigo member, joined on as artistic director, with Weathers, who's been leading the Reading Parlor series at CoHo, and Torres-Garner as associate artistic directors.
The company will present three productions for its inaugural season, with Anatomy of a Hug looking like a very promising start. The script, by Kat Ramburg, concerns an awkward, TV-obsessed young woman trying to cope with the unexpected attentions of a sweetly enthusiastic co-worker and with an uncomfortable reunion with her mother, released from prison as she's dying from cancer. Glick directs, with Jessica Hillebrand in the central role, plus Jacklyn Maddux, Murri Lazaroff-Babin and Shareen Jacobs.
February will bring Stephanie Alison Walker's Friends With Guns, with Torres-Garner directing Glick and Weathers, then sometime in the spring, Weathers will direct Torres-Garner and others in Rachel Bonds' Curve of Departure.
Regardless of the choice of the scripts and the talent on stage, fledgling theaters can have a rough go of it. Is this a scary thing to be doing?
"Not when I'm in the rehearsal hall, 'cause that's my jam," Glick says with a smile. But he's approaching it all seriously. "We have to run it like a business, not like a bunch of theater kids having fun. It's about not having egos but working together toward a collective goal. And I'm not in this for one season, My hope is that we're doing this for long-term prospects."
Photographing the craft and grit of winemaking
Zooming into a new theater
Looking Back 2020: Reports from the orchestra seats
The year in dance: 2020, go away!
Passages: The ones we lost in 2020
Patrick F. Smith: Recovering Old Portland
After the Fire 2: Starting Again
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text:
Government // Leadership
David F Carr
New York State Plans Expansion Of STEM Schools
Gov. Cuomo's budget proposal includes $4 million to establish 10 new Early College High School programs, similar to the IBM-backed P-TECH in New York City, aimed at preparation for high-tech jobs.
Educational 'Technology' Across the Ages
(click image for larger view and for slideshow)
New York Gov. Andrew Cuomo is proposing to establish 10 new programs modeled on Pathways in Technology Early College High School (P-TECH), the IBM-backed school in Brooklyn that takes students through high school and two years of college with the goal of meeting the demand for workers with the right skills for high tech jobs.
P-TECH is the same program President Obama singled out for praise in his State of the Union Address earlier this month. Gov. Cuomo's office announced the technology education partnership program Tuesday as part of his executive budget proposal.
Following the IBM blueprint for science, technology, engineering, and math (STEM) education, P-TECH and schools like it are starting to be established in other cities such as Chicago to help students earn an associate degree, which IBM considers the minimum qualification for an entry-level job in a technical discipline. IBM has promised that those who graduate from the program will be "first in line" for a job at IBM, whether they stop with the equivalent of a community college education or continue on to earn a four-year degree.
[ Want to 'THINK' like an IBMer? Read IBM Showcases STEM Support At Disney's Epcot. ]
New York's planned expansion of the program would establish one new school on this model in each of 10 economic development zones around the state. New corporate sponsors would also join IBM in offering students mentors from the business world, along with the same "first in line" consideration for a job after graduation. IBM will serve as the corporate partner for two of the new schools, will help recruit other private-sector partners, and will offer training to participating schools and mentors.
Gov. Cuomo's budget proposal includes $4 million in incentives for districts that establish a school on this model of public-private partnership.
IBM's Stan Litow, one of the architects of this education model, said he can't forecast how the governor's budget will fare with the legislature, but that this $4 million incentive payment ought to survive, particularly with all the interest generated by Obama's endorsement. Although many details remain to be worked out, school districts already are lining up to be part of the initiative, he said.
"I don't think interest from companies, interest from school districts, certainly interest from children and parents -- I don't think that's going to be the problem," said Litow, a former deputy chancellor of the NYC schools and IBM's VP for corporate citizenship and corporate affairs.
In a statement for the governor's press release, Dick Parsons, chair of the New NY Education Commission and senior advisor at Providence Equity Partners LLC, said, "Governor Cuomo recognizes that quality education is a game changer for New York's students and New York's economy. By expanding the P-TECH model across the State, as recommended by the New NY Education Reform Commission, Governor Cuomo and IBM will add fuel to local economies and businesses, while also providing innovative models of education reform that our public school system needs."
Litow said he had been encouraged by the strong recommendation from the commission chaired by Parsons, the former CEO of Time Warner, but was glad to see it translated into action. "This is a very big commitment and it demonstrates an enormous amount of leadership from the governor," Litow said.
He hopes to see the initiative mirrored by other states across the country. "The numbers really dictate the need for some sort of strong action," Litow said, referring to high community college dropout rates and the mismatch between employer needs and the skills of graduating students. "The problem is clear, and if we're interested in addressing the skills crisis we need a new model," he said.
Follow David F. Carr on Twitter @davidfcarr or Google+.
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PJS880,
re: New York State Plans Expansion Of STEM Schools
I think that this is a great idea and that this should have been implemented a long time ago in this state. Who would have though that teaching students actual real world issues and problem solving skills would be the foundation to a ideal future hire. I do not know what companies would not wan to sponsor this and have their pick of up and coming IT experts trained to handle the situations that are currently facing businesses. Instead hire people who can with little training after hire fall right into place. A perfect transition for the company and the new hire.
Paul Sprague
InformationWeek Contributor