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∈L1 or, more generally, when f∈W(L1,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.
ASJC Scopus subject areas
- Applied Mathematics