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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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##### References:
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