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On the Dirichlet kernels and a Hardy space with respect to the Vilenkin system. (English) Zbl 0577.42021
The authors study a family of complete orthonormal systems defined by N. Ya. Vilenkin [Izv. Akad. Nauk SSSR, Ser. Mat. 11, 363-400 (1947; Zbl 0036.356)]. They find some upper bounds for the Dirichlet kernel with respect to these systems and prove a Hardy type inequality as follows.
Theorem 4. There exists an absolute constant $$C>0$$ such that $$\sum^{\infty}_{k=1}k^{-1}| \hat f(k)| \leq C\| f\|$$ $$(f\in H(G_ m))$$, where $$\hat f($$k) are the Vilenkin-Fourier coefficients of the function f and $$H(G_ m)$$ is the set of all functions $$f=\sum^{\infty}_{i=0}\lambda_ ia_ i(x),$$ where $$\sum^{\infty}_{i=0}| \lambda_ i| <\infty$$ and the $$a_ i(x)\in L^{\infty}(G_ m)$$ are atoms, i.e. either $$a_ i(x)\equiv 1$$ or there is an interval I such that supp $$a_ i(x)\subset I$$, $$| a_ i(x)| \leq | I|^{-1}$$ and $$\int_{I}a_ i(x)dx=0$$ ($$| I|$$ denotes the Haar measure of I); $$\| f\| =\inf \sum^{\infty}_{i=0}| \lambda_ i|.$$
Reviewer: F.Móricz

##### MSC:
 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 42C30 Completeness of sets of functions in nontrigonometric harmonic analysis 43A17 Analysis on ordered groups, $$H^p$$-theory
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