Wiener amalgams and pointwise summability of Fourier transforms and Fourier series

Abstract

This paper provides a fairly general approach to summability questions for multi-dimensional Fourier transforms. It is based on the use of Wiener amalgam spaces $W(L_p,\ell_q)({\mathbb R}^d)$, Herz spaces and weighted versions of Feichtinger's algebra and covers a wide range of concrete special cases (20 of them are listed at the end of the paper). It is proved that under some conditions the maximal operator of the $\theta$-means $\sigma_T^\theta f$ can be estimated pointwise by the Hardy–Littlewood maximal function. From this it follows that $\sigma_T^\theta f \,{\to}\, f$ a.e. for all $f\in W(L_1,\ell_\infty)({\mathbb R}^d)$, hence $f\in L_p({\mathbb R}^d)$ for any $1\leq p\leq \infty$. Moreover, $\sigma_T^\theta f(x)$ converges to $f(x)$ at each Lebesgue point of $f\in L_1({\mathbb R}^d)$ (resp. $f\in W(L_1,\ell_\infty)({\mathbb R}^d)$) if and only if the Fourier transform of $\theta$ is in a suitable Herz space. In case $\theta$ is in a Besov space or in a weighted Feichtinger's algebra or in a Sobolev-type space then the a.e. convergence is obtained. Some sufficient conditions are given for $\theta$ to be in the weighted Feichtinger's algebra. The same results are presented for multi-dimensional Fourier series.

A sampling theory for non-decaying signals

“…We shall show that both the sampling and the reconstruction are stable through Riesz-type bounds, provided that the kernel ϕ is an element of an appropriate weighted hybrid-norm space [28,29]. These results are extensions of what has been presented in [10] where the kernel ϕ is required to be in a weighted Wiener amalgam space [30][31][32][33][34][35][36][37][38], which is not sensitive to the power p of the function spaces. By relaxing amalgam spaces to hybrid-norm spaces, we can control p which, together with the weighting function, dictates the order of growth of the signals.…”

“…Deeper results regarding Wiener amalgam spaces and their generalizations can be found in [33][34][35][36][37][38]. We note some obvious inclusions of hybrid-norm spaces:…”

Approximate identities on some homogeneous Banach spaces

“…General sufficient conditions for pointwise convergence of approximate identities have been obtained in [9] for a class of spaces equivalent to Stepanoff spaces (namely, Wiener amalgams); moreover, the spaces M p and M p 0 are equivalent to certain Herz spaces (respectively, inhomogeneous and homogeneous), where pointwise convergence of approximate identities holds by the results of [9] and [10]. …”

“…Remark 2.4 suggests that the same should be true for M p 0 . General sufficient conditions for pointwise convergence of approximate identities have been obtained in [9] for a class of spaces equivalent to Stepanoff spaces (namely, Wiener amalgams); moreover, the spaces M p and M p 0 are equivalent to certain Herz spaces (respectively, inhomogeneous and homogeneous), where pointwise convergence of approximate identities holds by the results of [9] and [10]. For the Marcinkiewicz spaces, instead, only a weak sufficient condition for pointwise convergence of approximate identities is known [2], based on a rather strong decay condition: counterexamples are available when this condition fails.…”

Herz spaces and summability of Fourier transforms

“…we [6] gave a sufficient and necessary condition for the θ-summability at each Lebesgue point of all f ∈ L p R d . As special cases of the Herz spaces we obtain the weighted L α p R d spaces.…”