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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.

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
Full Text: DOI
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