On double cosine, sine, and Walsh series with monotone coefficients

Abstract

We extend from one-dimensional to two-dimensional series the results by Hardy and Littlewood [6] on the L r {L^r} -integrability of the sum f f and the results by Stechkin [10] on the L 1 {L^1} -integrability of the maximum partial sum M ∗ {M^*} in the case of cosine and sine series with monotone coefficients. Among others, we prove that the L r {L^r} -integrability of f f and M ∗ {M^*} is essentially equivalent for r > 1 r > 1 in the two-dimensional setting, too. Simultaneously, we extend our earlier results in [7] from one-dimensional to two-dimensional Walsh series.