Almost everywhere convergence of Ciesielski-Fourier series of H1functions

Abstract

It is proved that a lacunary sequence of the Ciesielski-Fourier series of f ∈ H 1 converges almost everywhere to f . Introduction.Let s n f denote the nth partial sum of the Walsh-Fourier series of f ∈ L 1 . Ladhawala and Pankratz [4] proved that if f is in the dyadic Hardy space H 1 and (n i , i ∈ N) is a lacunary sequence of positive integers, then s n i f converges a.e. to f . Moreover, Schipp and Simon [6] verified that if (u) = o(log log u) (u → ∞) then there exists a function in H 1 (H 1 ) whose full sequence of partial sums diverges everywhere. Especially, if (u) = 1(u 1) then we get H 1 (H 1 ) = H 1 , i.e., it follows the existence of f ∈ H 1 such that s n f diverges everywhere (see Ladhawala and Pankratz [4]). The analogous results for trigonometric Fourier series can be found in Zygmund [17, II. p. 235] and for Vilenkin-Fourier series in Young [15].In this paper we extend these results to Ciesielski-Fourier series, which are generalizations of the Walsh-Fourier series. The Ciesielski systems can be obtained from the spline systems of order (m, k) in the same way as the Walsh system arises from the Haar system (see Ciesielski [2, 1]).

Results on spline-Fourier and Ciesielski-Fourier series

“…235] and for Vilenkin-Fourier series by Young [73]. Now we extend this result to Ciesielski-Fourier series (see Weisz [62]). …”