Fridli, S.; Simon, P. On the Dirichlet kernels and a Hardy space with respect to the Vilenkin system. (English) Zbl 0577.42021 Acta Math. Hung. 45, 223-234 (1985). 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 Cited in 1 ReviewCited in 6 Documents 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 Keywords:complete orthonormal systems; Dirichlet kernel; Vilenkin-Fourier coefficients Citations:Zbl 0036.356 PDFBibTeX XMLCite \textit{S. Fridli} and \textit{P. Simon}, Acta Math. Hung. 45, 223--234 (1985; Zbl 0577.42021) Full Text: DOI References: [1] S. V. Bockariev, The logaritmic growth of the aritmetic means of the Lebesgue functions with respect to bounded orthonormal systems,DAN SSSR,223 (1975), 16–19 (in Russian). [2] J. A. Chao, Hardy spaces on regular martingales, Martingale Theory in Harmonic Analysis and Banach Spaces, Cleveland, Ohio, 1981, 18–28.,Lecture Notes in Mathematics,939, Springer (Berlin). [3] J. D. Chen, A theorem of Kaczmarz on multiple generalized Walsh-Fourier series,Bull. Inst. Math. Acad. Sinica,10 (1982), 205–212. · Zbl 0517.42034 [4] R.R. Coifman, G. L. Weiss, Extensions of Hardy spaces and their use in analysis,Bull. Amer. Math. Soc.,83 (1977), 569–645. · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5 [5] B. I. Golubov, On a class of complete orthogonal systems,Sib. Mat. Z.,9 (1968), 297–314 (in Russian). · Zbl 0183.06301 [6] F. Móricz, Lebesgue functions and multiple function series I.,Acta Math. Acad. Sci. Hungar.,37 (1981), 481–496. · Zbl 0469.42009 · doi:10.1007/BF01895150 [7] P. Simon, Investigations with respect to the Vilenkin system,Annales Univ. Sci. Budapest (to appear). · Zbl 0586.43001 [8] N. Ja. Vilenkin, On a class of complete orthonormal systems,Izv. Akad. Nauk SSSR, ser. mat.,11 (1947), 363–400. · Zbl 0036.35601 [9] C. Watari, On generalized Walsh-Fourier series,Tohoku Math. J.,10 (1958), 211–241. · Zbl 0085.05803 · doi:10.2748/tmj/1178244661 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.