Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)
GÁT, G., On a theorem of type Hardy-Littlewood with respect to the Vilenkin-like systems
On a theorem of type Hardy-Littlewood. . . 85 We mention some examples. (k) 1. If a — 1 for each k.j E N, then we have the "ordinary" Vilenkin 3 systems. 2. If rrij = 2 for all j E N and a^'^ = where ßj(x) = exp (2TTZ + • • • + 2J?r)) (nj E N, * E G m), then we have the character system of the group of 2-adic integers (see e.g. [5], W). 3. If / / OO \ CO / n(x): = exp í 2tu I —j Y^ xJ MJ ) ( x e n E N). then we have a Vilenkin-Hke system which is usefull in the approximation theory of limit periodic, almost even arithmetical functions ([2], [4]). In [3] we proved that a Vilenkin-like system is orthonormal and complete in L l(Gm). Define the Fourier coefficients, the partial stuns of the Fourier series, the Dirichlet kernels with respect to the Vilenkin-like system X as follows. / n — 1 /X«, s*f = s nf-.= J£f x(k)xk, k= 0 n-1 D*(y,x) = D n(y,x)\=^Xn(y)Xn{x): k= 0 It is known ([2]) that / \ r^ / \ f ^n, if y - x £ ^n(O), D M n(y,x) = D M n(y-x)={^ / y. x^ I n\o)[ S MJ{y) = M n f /d/z = E nf(y) (/ E L\G m), n E N) and OO TTlj— 1 Dn{y, x) = Xn{y)Xn{x) ^ D M j (y - x) ^(x) j= 0 p=m ; —nj
