Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

BLAHOTA, I., Relation between Dirichlet kernels with respect to Vilenkin-like systems

114 István Blahota 2.2 Let n € {jM

t\t,j G N, m

t < j}. It is easy to see that x' - y' G I

t, but x' - y' £ I

t

+1. Then Df

M

t(x\y') = cDf

M

i(x'-y% where c ^ 1. We will prove that - y') / 0, thus * D1MM' - »')• We have by lemma - y') = DlSx' - y') X>M#,(*' - •) = M

t £ W - /) = TI /27TÍ\ 2 I-EXPFE)

2 M^exp — = M

t T^W

0' S! l-exp(^) since m

t J(j. The proof of theorem is complété. Acknowledgement The author wishes to thank to Professor G. Gát for setting the problem. References [1] G. GÁT, VILENKIN, Fourier Sériés and Limit Peridic Arithmetic Functions, Colloquia Mathematica SocietaÉis János Bolyai, 58 (1990), 315332. [2] G. GÁT, Orthonormal systems on Vilenkin groups, Acta Mathematica Hungarica 58 (1—2) (1991), 193-198 [3] F. SCHIPP, W. R. WADE, P. SIMON and J. PÁL, Walsh Sériés, An Introduction to Dyadic Harmonie Analysis, Akadémiai Kiadó, Budapest, and Adam Hilger, Bristol and New York (1990). [4] G. H. AGAJEV, N. YA. VILENKIN, G. M. DZSAFARLI, A. I. RUBINSTEIN, Multiplicative systems of funetyions and harmonie analysis on 0dimension al groups, Izd. "ELM" (Baku, SSSR) (1981).