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Summability of Gabor expansions and Hardy spaces. (English) Zbl 1221.42015
The paper deals with Wiener amalgam spaces \(W(X, \ell^q)\) on \({\mathbb R}^d\), defined by the condition \(\| f\|_{W(X,\ell^q)}=\sum_{k\in {\mathbb Z}^d}\| f_{|[k,k+1)}\|_X^q<\infty\), when \(X=L^p,L^{1,\infty}\) and \(h^p\) (the local Hardy space), and with the general \(\theta\)-summability method defined by a function \(\theta\) in the \(W(L^\infty,\ell^1)\)-closure of continuous functions which has an integrable Fourier transform \(\widehat\theta\), with some extra conditions on the derivatives of \(\widehat\theta\).
If \(\sigma_*^\theta h\) is the maximal function of the \(\theta\)-means for Gabor series, the author obtains boundedness results for \(\sigma_*^\theta :h_p\to L^p\) and \(\sigma_*^\theta :W(h^p,\ell^\infty)\to W(L^p,\ell^\infty)\), and then the a.e. convergence of the \(\theta\)-summation method for functions from \(W(L^1,\ell^\infty)\).
MSC:
42B08 Summability in several variables
42B35 Function spaces arising in harmonic analysis
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