Statistical Convergence of Walsh-fourier Series

This is a brief and concise account of the basic concepts and results on statistical convergence, strong Cesàro summability and Walsh-Fourier series. To emphasize the significance of statistical convergence, for example we mention the fact that the one-dimensional Walsh-Fourier series of an integrable (in Lebesgue’s sense) function may be divergent almost everywhere, but it is statistically convergent almost everywhere. The case of multi-dimensional Walsh-Fourier series is also considered. For future research, we raise two open problems and formulate two conjectures. 1. Statistical convergence of sequences. The concept of statistical convergence was first introduced and studied by Fast [2] in 1951. We note that in the first edition (1935) of the book “Trigonometric Series” by Anthony Zygmund one can find two theorems involving the concept of almost convergence (see [12, Vol. 2, on pp. 181 and 188]), which turned out to be equivalent to the concept of statistical convergence. A sequence (sk : k = 1, 2, . . .) of real or complex numbers is said to be statistically convergent to some finite number L, if for every ε > 0 we have (1.1) lim n→∞ n−1|{k ≤ n : |sk − L| > ε}| = 0, where by k ≤ n we mean k = 1, 2, . . . , n, and by |S| the cardinality of the set S ⊆ P, the set of positive integers. Clearly, the statistical limit L is uniquely determined. The following concept is due to Fridy [3]. A sequence (sk) is said to be statistically Cauchy if for every ε > 0 there exists an integer m such that lim n→∞ n−1|{k ≤ n : |sk − sm| > ε}| = 0; and he proved that (sk) is statistically convergent if and only if it is statistically Cauchy. The concept of statistical convergence can be reformulated in terms of natural density. To this end, we recall (see, e.g. [6, on p. 290]) that the natural (or asymptotic) density of a set S ⊆ P is defined by d(S) := lim n→∞ n−1|{k ≤ n : k ∈ S}|, 2000 Mathematics Subject Classification. Primary 42C10; Secondary 40A05, 40G05.