Formula omitted]-summability of higher-dimensional Fourier series

Abstract

It is proved that the maximal operator of the ℓ 1 -Fejér means of a d -dimensional Fourier series is bounded from the periodic Hardy space H p ( T d ) to L p ( T d ) for all d / ( d + 1 ) < p ≤ ∞ and, consequently, is of weak type (1, 1). As a consequence we obtain that the ℓ 1 -Fejér means of a function f ∈ L 1 ( T d ) converge a.e. to f . Moreover, we prove that the ℓ 1 -Fejér means are uniformly bounded on the spaces H p ( T d ) and so they converge in norm ( d / ( d + 1 ) < p < ∞ ) . Similar results are shown for conjugate functions and for a general summability method, called θ -summability. Some special cases of the ℓ 1 – θ -summation are considered, such as the Weierstrass, Picard, Bessel, Fejér, de la Vallée Poussin, Rogosinski and Riesz summations.