INEQUALITIES IN SUMMABILITY THEORY OF FOURIER SERIES

Some recent results on a general summability method, the so-called theta-summability, are summarized for one-dimensional Fourier series. Natural choices of theta are investigated, i.e., if theta is in Wiener amalgam spaces, Feichtinger's algebra or modulation spaces. Sufficient and necessary conditions are given for the uniform and L-1 norm and a e. convergence of the theta-means sigma(theta)(n)f to the function f. The maximal operator of the theta-means is investigated and it is proved that it is bounded on L-p spaces and on Hardy spaces.