an:05638437
Zbl 1185.42006
Weisz, Ferenc
Pointwise summability of Gabor expansions
EN
J. Fourier Anal. Appl. 15, No. 4, 463-487 (2009).
00255586
2009
j
42B08 42C15 42C40 42A38 46B15
Wiener amalgam spaces; Herz spaces; \(\theta\)-summability; Gabor expansions; Gabor frames; time-frequency analysis; Hardy--Littlewood inequality
Author's abstract: A general summability method, the so-called \(\theta\)-summability method is considered for Gabor series. It is proved that if the Fourier transform of \(\theta\) is in a Herz space then this summation method for the Gabor expansion of \(f\) converges to \(f\) almost everywhere when \(f \in L_1\) or, more generally, when \(f \in W(L_1, \ell_{\infty})\) (Wiener amalgam space). Some weak type inequalities for the maximal operator corresponding to the \(\theta\)-means of the Gabor expansion are obtained. Hardy--Littlewood type maximal functions are introduced and some inequalities are proved for these.
Richard A. Zalik (Auburn University)