This paper is concerned with the characterization of $α$-modulation spaces by Banach frames, i.e., stable and redundant non-orthogonal expansions, constituted of functions obtained by a suitable combination of translation, modulation and dilation of a mother atom. In particular, the parameter $α\in [0,1]$ governs the dependence of the dilation factor on the frequency. The result is achieved by exploiting intrinsic properties of localization of such frames. The well-known Gabor and wavelet frames arise as special cases ($α= 0$) and limiting case ($ α\to 1)$, to characterize respectively modulation and Besov spaces. This intermediate theory contributes to a further answer to the theoretical need of a common interpretation and framework between Gabor and wavelet theory and to the construction of new tools for applications in time-frequency analysis, signal processing, and numerical analysis.
We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.
We introduce and study a new class of translation-modulation invariant Banach spaces of ultradistributions. These spaces show stability under Fourier transform and tensor products; furthermore, they have a natural Banach convolution module structure over a certain associated Beurling algebra, as well as a Banach multiplication module structure over an associated Wiener-Beurling algebra. We also investigate a new class of modulation spaces, the Banach spaces of ultradistributions $\mathcal{M}^F$ on $\mathbb{R}^{d}$, associated to translation-modulation invariant Banach spaces of ultradistributi…
We study the space of tempered ultradistributions whose convolutions with test functions are all contained in a given translation-modulation invariant Banach space of ultradistributions. Our main result will be the first structural theorem for the aforementioned space. As an application we consider several extensions of convolution.
We deduce continuity, compactness and invariance properties for quasi-Banach Orlicz modulation spaces. We characterize such spaces in terms of Gabor expansions and by their images under the Bargmann transform.
We introduce new quasi-Banach modulation spaces on locally compact abelian (LCA) groups which coincide with the classical ones in the Banach setting and prove their main properties. Then we study Gabor frames on quasi-lattices, significantly extending the original theory introduced by Gröchenig and Strohmer. These issues are the key tools in showing boundedness results for Kohn-Nirenberg and localization operators on modulation spaces and studying their eigenfunctions' properties. In particular, the results in the Euclidean space are recaptured.
We prove the boundedness of a general class of multipliers and Fourier multipliers, in particular of the Hilbert transform, on quasi-Banach modulation spaces. We also deduce boundedness for multiplications and convolutions for elements in such spaces.
We deduce trace properties for modulation spaces of Gelfand-Shilov distributions. We use these properties to show that pseudo-differential operators with amplitudes in suitable modulation spaces, agree with pseudo-differential operators of normal type whose symbols belong to (other) modulation spaces.
We extend the Gabor analysis in \cite{GaSa} to a broad class of modulation spaces, allowing more general mixed quasi-norm estimates and weights in the definition of the modulation space quasi-norm. For such spaces we deduce invariance and embedding properties, and that the elements admit reconstructible sequence space representations using Gabor frames.
We introduce global wave-front sets $\operatorname{WF}_{\mathcal B} (f)$, $f\in {\mathscr S}^\prime(\textbf{R}^d)$, with respect to suitable Banach or Fréchet spaces ${\mathcal B}$. An important special case is given by the modulation spaces ${\mathcal B}=M(ω,\mathscr B)$, where $ω$ is an appropriate weight function and $\mathscr B$ is a translation invariant Banach function space. We show that the standard properties for known notions of wave-front set extend to $\operatorname{WF}_{\mathcal B} (f)$. In particular, we prove that micro locality and microellipticity hold for a class of globally …
We deduce continuity properties for pseudo-differential operators with symbols in quasi-Banach Orlicz modulation spaces when rely on other quasi-Banach Orlicz modulation spaces. In particular we extend certain results in \cite{GH1,GH2,Toft2,Toft10,Toft16}.
We establish continuity and Schatten-von Neumann properties for matrix operators with matrices satisfying mixed quasi-norm estimates. These considerations also include the case when the Lebesgue and Schatten parameters are allowed to stay between $0$ and $1$. We use the results to deduce continuity and Schatten-von Neumann properties for pseudo-differential operators with symbols in a broad class of modulation spaces.
We prove modulation invariant embedding bounds from Bochner spaces $L^p(\mathbb{W};X)$ on the Walsh group to outer-$L^p$ spaces on the Walsh extended phase plane. The Banach space $X$ is assumed to be UMD and sufficiently close to a Hilbert space in an interpolative sense. Our embedding bounds imply $L^p$ bounds and sparse domination for the Banach-valued tritile operator, a discrete model of the Banach-valued bilinear Hilbert transform.
We obtain Gabor frame characterisations of modulation spaces defined via a class of translation-modulation invariant Banach spaces of distributions that was recently introduced in $[10]$. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterised by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalises several fundamental results for the classical modulation spaces $ M^{p,q}_{w}$. Due to the absence of solidity assumptions on the Banach spaces defining these modul…
In this paper, we consider the trace theorem for modulation spaces, alpha modulation spaces and Besov spaces. For the modulation space, we obtain the sharp results.
We provide sequence space representations for the test function space $\mathcal{D}_{E}$ and the distribution space $\mathcal{D}^{\prime}_{E}$ associated to a Banach space $E$ belonging to a broad class of translation-modulation invariant Banach spaces of distributions. The spaces $\mathcal{D}_{E}$ and $\mathcal{D}^{\prime}_{E}$ generalize the classical Schwartz spaces $\mathcal{D}_{L^p}$ and $\mathcal{D}^{\prime}_{L^p}$, respectively. Our proof is based on Gabor frame characterizations of $\mathcal{D}_{E}$ and $\mathcal{D}^{\prime}_{E}$, which are also established here and are of independent i…
We generalize the classical coorbit space theory developed by Feichtinger and Gr"ochenig to quasi-Banach spaces. As a main result we provide atomic decompositions for coorbit spaces defined with respect to quasi-Banach spaces. These atomic decompositions are used to prove fast convergence rates of best $n$-term approximation schemes. We apply the abstract theory to time-frequency analysis of modulation spaces $M^{p,q}_m$, $0< p,q \leq \infty$.
Let (e_i) be a dictionary for a separable Banach space X. We consider the problem of approximation by linear combinations of dictionary elements with quantized coefficients drawn usually from a `finite alphabet'. We investigate several approximation properties of this type and connect them to the Banach space geometry of X. The existence of a total minimal system with one of these properties, namely the coefficient quantization property, is shown to be equivalent to X containing c_0.
We prove that the class of trilinear multiplier forms with singularity over a one dimensional subspace, including the bilinear Hilbert transform, admit bounded $L^p$-extension to triples of intermediate $\mathrm{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\mathrm{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\mathrm{UMD}$-valued setting. This is then employed to obtain appropriate single tree estimates by appealing to the $\mathrm{UMD}$-valued bound for bilinea…
We identify the modulation spaces associated to tensor products of amalgam spaces having a large class of Banach spaces as their local component. As consequences of the main results, we describe the modulation spaces associated to tensor products of various $L^p$ spaces.
We investigate mapping properties for the Bargmann transform on modulation spaces whose weights and their reciprocals are allowed to grow faster than exponentials. We prove that this transform is isometric and bijective from modulation spaces to convenient Lebesgue spaces of analytic functions. We use this to prove that such modulation spaces fulfill most of the continuity properties which are well-known when the weights are moderated. Finally we use the results to establish continuity properties of Toeplitz and pseudo-differential operators in the context of these modulation spaces.
Let $UC(K)$ denote the Banach space of all bounded uniformly continuous functions on a hypergroup $K$. The main results of this article concern on the $α$-amenability of $UC(K)$ and quotients and products of hypergroups. It is also shown that a Sturm-Liouville hypergroup with a positive index is $α$-amenable if and only if $α=1$.
We study decay and smoothness properties for eigenfunctions of compact localization operators. Operators with symbols a in the wide modulation space M^{p,\infty} (containing the Lebesgue space L^p), p<\infty, and windows \f_1,\f_2 in the Schwartz class are known to be compact. We show that their L^2-eigenfuctions with non-zero eigenvalues are indeed highly compressed onto a few Gabor atoms. Similarly, for symbols a in the weighted modulation spaces M^{\infty}_{v_s\otimes 1} (\rdd), s>0 (subspaces of M^{p,\infty}(\rdd), p>2d/s) the L^2-eigenfunctions of the localization operator are actually Sc…
We prove that if $ω_1$ and $ω_2$ are moderate weights and $\mascB$ is a suitable (quasi-)Banach function space, then a necessary and sufficient condition for the embedding $i\, :\, M (ω_1,\mascB )\to M (ω_2,\mascB )$ between two modulation spaces to be compact is that the quotient $ω_2/ω_1$ vanishes at infinity. Moreover we show, that the boundedness of $ω_2/ω_1$ a necessary and sufficient condition for the previous embedding to be continuous.
A classical result of Cembranos and Freniche states that the C(K, X) spaces contains a complemented copy of c_0 whenever K is an infinite compact Hausdorff space and X is an infinite dimensional Banach space. This paper takes this result as a starting point and begins a study of the conditions under which the spaces C(alpha), alpha<omega_1 are quotients of or complemented in spaces C(K,X). In contrast to the c_0 result, we prove that if C(beta mathbb N times [1,omega], X) contains a complemented copy of C(omega^omega) then X contains a copy of c_0. Moreover, we show that C(omega^omega) is not …
We give a self-contained introduction to (quasi-)Banach modulation spaces of ultradistributions, and review results on boundedness for multiplications and convolutions for elements in such spaces. Furthermore, we use these results to study the Gabor product. As an example, we show how it appears in a phase-space formulation of the nonlinear cubic Schrödinger equation.
This paper is concerned with frame decompositions of $α$-modulation spaces. These spaces can be obtained as coorbit spaces for square-integrable representations of the affine Weyl-Heisenberg group modulo suitable subgroups. The theory yields canonical constructions for Banach frames or atomic decompositions in $α$-modulation spaces. A necessary ingredient in this abstract machinery is the existence of generating functions that are admissible for the representation. For numerical purposes, admissible atoms with compact support are necessary. We show new admissibility conditions that considerabl…
We show continuity properties for the pseudo-differential operator $\operatorname{Op} (a)$ from $M(ω_0ω,\mathscr B )$ to $M(ω,\mathscr B )$, for fixed $s,σ\ge 1$, $ω,ω_0\in \mathscr P _{s,σ}^0$ ($ω,ω_0\in \mathscr P _{s,σ}$), $a\in Γ^{σ,s}_{(ω_0)}$ ($a\in Γ^{σ,s;0}_{(ω_0)}$) , and $\mathscr B$ is an invariant Banach function space.
Let $s\ge 1$, $ω,ω_0\in \mathscr P_{E,s}^0$, $a\in Γ_{s}^{(ω_0)}$, and let $\mathscr B$ be a suitable invariant quasi-Banach function space, Then we prove that the pseudo-differential operator $\operatorname{Op} (a)$ is continuous from $M(ω_0ω,\mathscr B )$ to $M(ω,\mathscr B )$.
The $α$-modulation transform is a time-frequency transform generated by square-integrable representations of the affine Weyl-Heisenberg group modulo suitable subgroups. In this paper we prove new conditions that guarantee the admissibility of a given window function. We also show that the generalized coorbit theory can be applied to this setting, assuming specific regularity of the windows. This then yields canonical constructions of Banach frames and atomic decompositions in $α$-modulation spaces. In particular, we prove the existence of compactly supported (in time domain) vectors that are a…
We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V.~Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into $(L^2(\mathbb{R}), ||\cdot{}||_2)$, nor is the Hilbert space structure …
$\newcommand{mc}[1]{\mathcal{#1}}$ $\newcommand{D}{\mc{D}(\mc{Q},L^p,\ell_w^q)}$ We present a framework for the construction of structured, possibly compactly supported Banach frames and atomic decompositions for decomposition spaces. Such a space $\D$ is defined using a frequency covering $\mc{Q}=(Q_i)_{i\in I}$: If $(φ_i)_{i}$ is a suitable partition of unity subordinate to $\mc{Q}$, then $\Vert g\Vert_{\D}:=\left\Vert\left(\Vert\mc{F}^{-1}(φ_i\hat{g})\Vert_{L^p}\right)_{i}\right\Vert_{\ell_w^q}$. We assume $\mc{Q}=(T_iQ+b_i)_{i}$, with $T_i\in{\rm GL}(\Bbb{R}^d),b_i\in\Bbb{R}^d$. Given a pr…
About twenty years ago Johnson and Zippin showed that every separable L_1(mu)-predual was isometric to a quotient of C(Delta ), where Delta is the Cantor set. In this note we will show that the natural analogue of the theorem for l_1-preduals does not hold. We will show that there are many l_1-preduals which are not isometric to a quotient of any C(K)-space with K a countable compact metric space. We also prove some general results about the relationship between l_1-preduals and quotients of C(K)-spaces with K a countable compact metric space. The results in this paper were presented at the Wo…
Gabor frames with Hermite functions are equivalent to sampling sequences in true Fock spaces of polyanalytic functions. In the L^2-case, such an equivalence follows from the unitarity of the polyanalytic Bargmann transform. We will introduce Banach spaces of polyanalytic functions and investigate the mapping properties of the polyanalytic Bargmann transform on modulation spaces. By applying the theory of coorbit spaces and localized frames to the Fock representation of the Heisenberg group, we derive explicit polyanalytic sampling theorems which can be seen as a polyanalytic version of the lat…
In the present investigation we are linking noncommutative geometry over noncommutative tori with Gabor analysis, where the first has its roots in operator algebras and the second in time-frequency analysis. Therefore we are in the position to invoke modern methods of operator algebras, e.g. topological stable rank of Banach algebras, to exploit the deeper properties of Gabor frames. Furthermore we are able to extend results due to Connes and Rieffel on projective modules over noncommutative tori to Banach algebras, which arise in a natural manner in Gabor analysis. The main goal of this inves…
We consider time-frequency localization operators $A_a^{φ_1,φ_2}$ with symbols $a$ in the wide weighted modulation space $ M^\infty_{w}(\mathbb{R}^{2d})$, and windows $ φ_1, φ_2 $ in the Gelfand-Shilov space $\mathcal{S}^{\left(1\right)}(\mathbb{R}^{d})$. If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of $A_a^{φ_1,φ_2}$ have appropriate subexponential decay in phase space, i.e. that they belong to the Gefand-Shilov space $ \mathcal{S}^{(γ)} (\mathbb{R}^{d}) $, where the parameter $γ\geq 1 $ is related to the growth of the considered weight. An im…
The classical criterion for compactness in Banach spaces of functions can be reformulated into a simple tightness condition in the time-frequency domain. This description preserves more explicitly the symmetry between time and frequency than the classical conditions. The result is first stated and proved for L^2(R^d), and then generalized to coorbit spaces. As special cases, we obtain new characterizations of compactness in Besov-Triebel-Lizorkin spaces, modulation spaces and Bargmann-Fock spaces.
This work represents a first systematic attempt to create a common ground for semi-classical and time-frequency analysis. These two different areas combined together provide interesting outcomes in terms of Schrödinger type equations. Indeed, continuity results of both Schrödinger propagators and their asymptotic solutions are obtained on $\hbar$-dependent Banach spaces, the semi-classical version of the well-known modulation spaces. Moreover, their operator norm is controlled by a constant independent of the Planck's constant $\hbar$. The main tool in our investigation is the joint applicatio…
