The modulus of continuity and the best approximation over the dyadic group (Q1207394)

The connection between the modulus of continuity and the best approximation of functions by Walsh polynomials was studied by \textit{C. Watari} [Tohoku Math. J., II. Ser. 15, 1-5 (1963; Zbl 0111.265)] for \(L^ p\) space, \(1\leq p<\infty\). A similar result for \(0<p<1\) was obtained by \textit{E. A. Storozenko}, \textit{V. G. Krotov} and \textit{P. Oswal'd} [Math. Sb., n. Ser. 98(140), 395-415 (1975; Zbl 0314.41004)]. On the other hand, direct and converse theorems for the Hardy space \(H^ p\), \(0<p<\infty\), over the \(n\)-dimensional torus were proved by \textit{L. Colzani} [Ann. Math. Pure Appl., IV. Ser. 137, 207-215 (1984; Zbl 0558.41017)]. In this paper these results for the \(H^ p\) space, \(0<p\leq 1\) and VMO space over the dyadic group are proved.