Inequalities in summability theory of Fourier series

Abstract

Some recent results on a general summability method, the so-called θ -summability, are summarized for one-dimensional Fourier series. Natural choices of θ are investigated, i.e., if θ is in Wiener amalgam spaces, Feichtinger's algebra or modulation spaces. Sufficient and necessary conditions are given for the uniform and L1 norm and a.e. convergence of the θ - means σ θ n f to the function f . The maximal operator of the θ -means is investigated and it is proved that it is bounded on Lp spaces and on Hardy spaces.