On planar Beurling and Fourier transforms

Beurling algebra analogues of the classical theorems of Wiener and Levy on absolutely convergent Fourier series

Let $f$ be a continuous function on the unit circle $Γ$, whose Fourier series is $ω$-absolutely convergent for some weight $ω$ on the set of integers $\mathcal{Z}$. If $f$ is nowhere vanishing on $Γ$, then there exists a weight $ν$ on $\mathcal{Z}$ such that $1/f$ had $ν$-absolutely convergent Fourier series. This includes Wiener's classical theorem. As a corollary, it follows that if $ϕ$ is holomorphic on a neighbourhood of the range of $f$, then there exists a weight $χ$ on $\mathcal{Z}$ such that \hbox{$ϕ\circ f$} has $χ$-absolutely convergent Fourier series. This is a weighted analogue of …

截断 Beurling 算子的加权 Sobolev 估计

Weighted Sobolev estimates of the truncated Beurling operator

本文证明了在给定条件下截断 Beurling 变换是加权 Sobolev 空间上的有界算子，并得出了其他 Cauchy 型积分的加权 Sobolev 估计。

Oscillation of Fourier Integrals with a spectral gap

Suppose that Fourier transform of a function f is zero on the interval [-a,a]. We prove that the lower density of sign changes of f is at least a/pi, provided that f is a locally integrable temperate distribution in the sense of Beurling, with non-quasianalytic weight. We construct an example showing that the last condition cannot be omitted.

Planar sampling sets for the short-time Fourier transform

This paper considers the problem of restricting the short-time Fourier transform to domains of nonzero measure in the plane and studies sampling bounds of such systems. In particular, we give a quantitative estimate for the lower sampling bound in the case of Hermite windows and derive a sufficient condition for a large class of windows in terms of a certain planar density. On the way, we prove a Remez-type inequality for polyanalytic functions. MSC2010: 42C40, 46E15, 46E20, 42C15

Interpolation in $\hat{\E^\prime}(\R)$

We give a geometric description of the interpolating varieties for the algebra of Fourier transforms of distributions (or Beurling ultradistributions) with compact support on the real line.

Wavelet frames, Bergman spaces and Fourier transforms of Laguerre functions

The Fourier transforms of Laguerre functions play the same canonical role in wavelet analysis as do the Hermite functions in Gabor analysis. We will use them as analyzing wavelets in a similar way the Hermite functions were recently by K. Groechenig and Y. Lyubarskii in "Gabor frames with Hermite functions, C. R. Acad. Sci. Paris, Ser. I 344 157-162 (2007)". Building on the work of K. Seip, "Beurling type density theorems in the unit disc, Invent. Math., 113, 21-39 (1993)", concerning sampling sequences on weighted Bergman spaces, we find a sufficient density condition for constructing frames …