The Segal algebra S0(Rd) and norm summability of Fourier series and Fourier transforms

A general summability method, the so-called theta-summability is considered for multi-dimensional Fourier series. Equivalent conditions are derived for the uniform and L (1)-norm convergence of the theta-means sigma (theta)(n)f to the function f. If f is in a homogeneous Banach space, then the preceeding convergence holds in the norm of the space. In case theta is an element of Feichtinger's Segal algebra S-0(R-d), then these convergence results hold. Some new sufficient conditions are given for theta to be in S-0(R-d). A long list of concrete special cases of the theta-summation is listed. The same results are also provided in the context of Fourier transforms, indicating how proofs have to be changed in this case.