Abstract
A general summability method, the so-called θ-summability method is considered for Gabor series. It is proved that if the Fourier transform of θ is in a Herz space then this summation method for the Gabor expansion of f converges to f almost everywhere when f∈L 1 or, more generally, when f∈W(L 1,ℓ ∞) (Wiener amalgam space). Some weak type inequalities for the maximal operator corresponding to the θ-means of the Gabor expansion are obtained. Hardy-Littlewood type maximal functions are introduced and some inequalities are proved for these.
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Communicated by A.J.E.M. Janssen.
This research was supported by the Hungarian Scientific Research Funds (OTKA) No. K67642.
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Weisz, F. Pointwise Summability of Gabor Expansions. J Fourier Anal Appl 15, 463–487 (2009). https://doi.org/10.1007/s00041-008-9046-9
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DOI: https://doi.org/10.1007/s00041-008-9046-9
Keywords
- Wiener amalgam spaces
- Herz spaces
- θ-summability
- Gabor expansions
- Gabor frames
- Time-frequency analysis
- Hardy-Littlewood inequality