The Segal Algebra ${\bf S}_0({\Bbb R}^d)$ and Norm Summability of Fourier Series and Fourier Transforms

Abstract

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier series. Equivalent conditions are derived for the uniform and L1-norm convergence of the θ-means σnθf 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 θ is an element of Feichtinger’s Segal algebra \({\bf S}_0({\Bbb R}^d)\), then these convergence results hold. Some new sufficient conditions are given for θ to be in \({\bf S}_0({\Bbb R}^d)\). A long list of concrete special cases of the θ-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.