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

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier series. Equivalent conditions are derived for the uniform and L ₁-norm convergence of the θ-means σ _n ^θ 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 θ is an element of Feichtinger’s Segal algebra , then these convergence results hold. Some new sufficient conditions are given for θ to be in . 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. This research was supported by Lise Meitner fellowship No M733-N04 and the Hungarian Scientific Research Funds (OTKA) No T043769, T047128, T047132.