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Pointwise summability of Gabor expansions. (English) Zbl 1185.42006
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.

MSC:
42B08 Summability in several variables
42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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