Pointwise Summability of Gabor Expansions

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 faL (1) or, more generally, when faW(L (1),a"" (a)) (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.