On the modulus of continuity with respect to functions defined on Vilenkin groups (Q1061338)

We generalize a theorem of \textit{A. I. Rubinshtejn} [Mat. Zametki 23, 379- 388 (1978; Zbl 0405.43006)]. By the usual definition the modulus of continuity of a function defined on a Vilenkin group, \(G_ m\), can be represented by a sequence of real numbers. Rubinshtejn characterized the sequences which turn to be the modulus of continuity of a function in \(C(G_ m)\), \(L^ 1(G_ m)\), \(L^ 2(G_ m)\). Now we prove his conjecture, namely the theorem is true for \(L^ p(G_ m)\) \((1\leq p<\infty)\) too.