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GÁT, G., On a theorem of type Hardy-Littlewood with respect to the Vilenkin-like systems

On a theorem of type Hardy-Littlewood. . . 89 as n —• oo, where constant r £ N is given as < 1 and n

0 —> 00 (as n —> 00) provided that r < n/riQ. That is the proof of the theorem is complete. References [1] AGAEV, G. H., VILENKIN, N. JA., DZHAFARLI, G. M., RUBINSTEIN, A. I., Multiplicative systems of functions and harmonic analysis on O-dimensional groups (in Russian), Izd. ("ELM"), Baku, 1981. [2] GÁT, G., Vilenkin-Fourier series and limit periodic arithmetic functions, Colloq. Math. Soc. János Bolyai 58, Approx. Theory, Kecskemét, Hungary, 1990, 316-332. [3] GAT, G., Orthonormal systems on Vilenkin groups, Acta Math. Hungar.. 58 (1-2) (1991), 193-198. [4] GÁT, G., On almost even arithmetical functions via orthonormal systems on Vilenkin groups, Acta Arith., 49 (2) (1991), 105-123. [5] HEWITT, E., ROSS, K., Abstract Harmonic Analysis, SpringerVerlag, Heidelberg, 1963. [6] SCHIPP, F., WADE, W. R., SIMON, P., PÁL, J., Walsh series, Introduction to dyadic harmonic analysis, Adam Hilger, Bristol and New York, 1990. [7] VILENKIN, N. JA

., On a class of complete orthonormal systems (in Russian), Izv. Akad. Na.uk. SSSR, Ser. Math. 11 (1947), 363 400. [8] YANO, S., A convergence test for Walsh-Fourier series, Tohoku Math. J., 6 (2-3) (1954), 226-230. BESSENYEI COLLEGE DEPARTMENT OF MATHEMATICS NYÍREGYHÁZA, P.O. Box 166. H-440Ü HUNGARY E-mail: gatgyiny2.bgytl.hu