Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)
GÁT, G., On a norm convergence theorem with respect to the Vilenkin system in the Hardy spaces
104 György Gát It is well-known that SMJ(X) = M N / f = E nf(x) (/ G L L(G M),NE N) oo rrij — 1 p—mj — ríj (x G G m,n G N, / G L 1(G r m)). For more détails on Vilenkin systems see In 1983. B. Smith proved ([7]) for the trigonométrie system the following convergence theorem for functions / in the "classical" Hardy Space. In 1987. P. Simon proved [6] this theorem for the Walsh system. (The Walsh system is a Vilenkin system, rrij = 2 for ail j G N in this case.) In 1993. the author improved [3] this resuit, that is proved this theorem for the Vilenkin systems on bounded Vilenkin groups and in the case of the H(G m ) "atomic" Hardy space — for unbounded ones, too. Does this theorem hold in the case of unbounded Vilenkin groups and the "maximal" Hardy space? (For unbounded Vilenkin groups H ^ H 1.) We give a negative answer for this question. Theorem. K sup n = OO, then there exists a fonction / G e.g. [1]. H 1 (G such that PROOF. Let (M k+ 1, x G 4(0,1) fk(x) : = < -M k+ l, xel k(0,A k) y 0, otherwise,
