Relation between Dirichlet kernels with respect to Vilenkin-like systems. (English) Zbl 0882.42017

Let \( m:= (m_k , k\in N) ( N:= \{ 0,1,\dots \} )\) be a sequence of integers not less than \(2\). Let \(Z_{m_k}\) be the \(m_k\)-th discrete cyclic group with Haar measure defined so that the measure of a singleton is \(1/m_k (k\in N).\) Let \(G_m := \prod_{k=0}^{\infty}Z_{m_k}\) with the product topology and measure. The system \((\psi _n : n\in N)\) is the character system of \(G_m\) called a Vilenkin system [see G. N. Agaev, N. Ya. Vilenkin, G. M. Dzhafarli, and A.I. Rubinshtejn, “Multiplicative systems of functions and harmonic analysis on zero-dimensional groups” (in Russian) (1981; Zbl 0588.43001)] or F. Schipp, W. R. Wade, and {P. Simon} [“Walsh series. An introduction to dyadic harmonic analysis” (1990; Zbl 0727.42017)]. The Vilenkin-like systems (denoted by \(\chi\)) were introduced by G. Gát [Acta Math. Hung. 58, No. 1/2, 193-198 (1991; Zbl 0753.11027)].
The author proves a statement on the Dirichlet kernel functions with respect to Vilenkin-like systems. Formerly, G. Gát [Approximation Theory, Proc. Conf., Kecskemét/Hung. 1990, Colloq. Math. Soc. János Bolyai 58, 315-332 (1991; Zbl 0760.42013)] gave a formula for the Dirichlet kernel \(D^{\chi}_n(x,y)\) which in general differs from \(D^{\psi}_n(x-y)\). The author discusses under what conditions \(D^{\chi}_n(x,y) = D^{\psi}_n(x-y)\) holds for all Vilenkin-like systems \(\chi\).


42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
43A75 Harmonic analysis on specific compact groups