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
On a norm convergence theorem with respect to the Vilenkin. . . 103 It is known that if the sequence m is bounded, then II 1 = H , otherwise H is a proper subset of H 1 [2]. Let Mo : = 1, M n+i := m nM n (n G N). Then each natural number n can be uniquely expressed as oo n i=o = J2 ni Mi ( ni ^ {0,1,m t- - 1}, i G N), where only a finite number of n^s difFer from zéro. The generalized Rademacher functions are defined as r n(x) exp ( 2iri —— ) (x G G m, n G N, i := \/—T) V ^n / Then oo j=o the nth Vilenkin fonction. The system ip : = : N G N) is called a Vilenkin system. Each ip n is a character of G m and ail the characters of G m are of this form. Define the m -adic addition as k © n\- + nj(moàmj))Mj (k,n G N). j=o Then, 1p k® n = 1p k1pn, Í>n(x + y) = ^nOO^nfa), ^n(-z) = = G G G m). Define the Fourier coefficients, the partial sums of the Fourier sériés, the Dirichlet kernels with respect to the Vilenkin system ip as follows ~ n — 1 f(n):= fï>n,S nf f(k)ip k, J G m i n k= 0 n — 1 D n(y,x) = D n(y - x): = k=o Then (Snf)(y) = [ f{x)D n(y - x)dx (n G N, y G G m, / G i 1^)). JG^
