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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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