Inclusion relations between [formula omitted]-Sobolev and Wiener amalgam spaces

Remarks on Fourier multipliers and applications to the wave equation

Metaplectic representation on Wiener amalgam spaces and applications to the Schrödinger equation

Invertibility of the Gabor frame operator on the Wiener amalgam space

Characterization of inclusion relations between Wiener amalgam and some classical spaces

Norm estimates for τ-pseudodifferential operators in Wiener amalgam and modulation spaces

We give a complete characterization of the continuity of pseudodifferential operators with symbols in modulation spaces M p, q , acting on a given Lebesgue space L r . Namely, we find the full range of triples (p, q, r), for which such a boundedness occurs. More generally, we completely characterize the same problem for operators acting on Wiener amalgam space W(L r , L s ) and even on modulation spaces M r, s . Finally, the action of pseudodifferential operators with symbols in is also investigated.

Let \(L^{q ( x ) } (\mathbb{R} ) \) be variable exponent Lebesgue space and \(\ell^{ \{ q_{n} \} }\) be discrete analog of this space. In this work we define the amalgam spaces W(L p(x),L q(x)) and \(W ( L^{p ( x ) },\ell^{ \{ q_{n} \} } ) \), and discuss some basic properties of these spaces. Since the global components \(L^{q ( x ) } (\mathbb{R} ) \) and \(\ell^{ \{ q_{n} \} }\) are not translation invariant, these spaces are not a Wiener amalgam space. But we show that there are similar properties of these spaces to the Wiener amalgam spaces. We also show that there is a variable exponent q(x) such that the sequence space \(\ell^{ \{ q_{n} \}}\) is the discrete space of \(L^{q ( x ) } (\mathbb{R} )\). By using this result we prove that \(W ( L^{p ( x ) },\ell ^{ \{ p_{n} \} } ) =L^{p ( x ) } (\mathbb{R} ) \). We also study the frame expansion in \(L^{p ( x ) } ( \mathbb{R} )\). At the end of this work we prove that the Hardy–Littlewood maximal operator from \(W ( L^{s ( x ) },\ell^{ \{t_{n} \} } ) \) into \(W ( L^{u ( x ) },\ell^{ \{v_{n} \} } ) \) is bounded under some assumptions.

Local Hardy spaces and summability of Fourier transforms

Wiener amalgam spaces with respect to orlicz spaces on the affine group

We establish existence of solutions in a scale of classes weaker than the finite energy Leray class and stronger than the infinite energy Lemarie-Rieusset class. The new classes are based on the $L...

Wiener amalgam spaces of quasianalytic ultradistributions

Let ? be a function in the Wiener amalgam space $\emph{W}_{\infty}(L_1)$ with a non-vanishing property in a neighborhood of the origin for its Fourier transform $\widehat{\phi}$ , ${\bf \tau}=\{\tau_n\}_{n\in {{\mathbb Z}}}$ be a sampling set on ? and $V_\phi^{\bf \tau}$ be a closed subspace of $L_2(\hbox{\ensuremath{\mathbb{R}}})$ containing all linear combinations of ?-translates of ?. In this paper we prove that every function $f\in V_\phi^{\bf \tau}$ is uniquely determined by and stably reconstructed from the sample set $L_\phi^{\bf \tau}(f)=\Big\{\int_{\hbox{\ensuremath{\mathbb{R}}}} f(t) \overline{\phi(t-\tau_n)} dt\Big\}_{n\in {{\mathbb Z}}}$ . As our reconstruction formula involves evaluating the inverse of an infinite matrix we consider a partial reconstruction formula suitable for numerical implementation. Under an additional assumption on the decay rate of ? we provide an estimate to the corresponding error.

In signal processing, discrete convolutions are usually involved in fast calculating coefficients of time-frequency decompositions like wavelet and Gabor frames. Depending on the regularity of the mother analyzing functions, one wants to detect the right resolution in order to achieve good approximations of coefficients. Local–global conditions on functions in order to get the convergence rate of Riemann-type sums to their scalar products in L 2 are presented. Wiener amalgam spaces, in particular for the space-time domain and W(L 2,l 1) for the frequency domain, give natural norms in order to estimate errors. In particular, relations between the rate of convergence of these series to integrals by increasing resolution and the (minimal) required Besov regularity are presented by means of functional and harmonic analysis techniques.

We define the grand Wiener amalgam space by using the classical Wiener amalgam space and the generalized grand Lebesgue space. Moreover we study the inclusions between these spaces and some applications. Finally we prove the Hölder inequality for the grand Wiener amalgam space. We also find the associate space and dual of this space, and we prove that the grand Wiener amalgam space is not reflexive.

A general summability method, the so called θ-summability is considered for multi-dimensional Fourier series, where θ is in the Wiener algebraW(C,ℓ1)($\mathbb{R}$d). It is based on the use of Wiener amalgam spaces, weighted Feichtinger's algebra, Herz and Hardy spaces. Under some conditions on θ, it is proved that the maximal operator of the θ-means is bounded from theHpHardy space toLp(orHp). This implies some norm and almost everywhere convergence results for the θ-means. Large number of special cases of the θ-summation are considered.