Herz spaces and restricted summability of Fourier transforms and Fourier series

A general summability method, the so-called θ-summability is considered for multi-dimensional Fourier transforms and Fourier series. A new inequality for the Hardy–Littlewood maximal function is verified. It is proved that if the Fourier transform of θ is in a Herz space, then the restricted maximal operator of the θ-means of a distribution is of weak type (1, 1), provided that the supremum in the maximal operator is taken over a cone-like set. From this it follows that σθ T f →f over a cone-like set a.e. for all f ∈ L1(Rd ). Moreover, σθ T f (x) converges to f (x) over a cone-like set at each Lebesgue point of f ∈ L1(Rd ) if and only if the Fourier transform of θ is in a suitable Herz space. These theorems are extended to Wiener amalgam spaces as well. The Riesz and Weierstrass summations are investigated as special cases of the θ-summation. © 2008 Elsevier Inc. All rights reserved