## Strong convergence theorem for Vilenkin-Fourier series.(English)Zbl 0987.42022

Let $$m=(m_0,m_1,...,m_k,...)$$ be a sequence of natural numbers such that $$m_k\geq 2, G_m$$ be the complete direct product of $$Z_{m_k}$$, and $$\|f\|_p= (\int_{G_m}|f|^pd\mu)^{1/p}, d\mu$$ be the Haar measure on $$G_m$$. Let $$(\psi_n)_{n=0}^\infty$$ be the character system on $$G_m$$, $$S_n(f)$$ be an $$n$$-th partial sum of $$f$$ with respect to $$(\psi_n)^\infty_{n=0}$$. If $$f^\ast$$ is the maximal martingale function of $$f$$, then the Hardy space $$H^p$$ with $$p>0$$ consists of all $$f$$ with $$\|f\|_{H^p}= \|f^\ast\|_p < \infty$$. Theorem. Let $$1/2<p<1$$ and let $$m$$ be arbitrary. Then there exists a constant $$C_p$$ such that for all $$f\in H^p$$ $\sum^\infty_{k=1} \|S_k(f)\|^p_pk^{p-2}\leq C_p\|f\|^p_{H^p}.$ If $$m$$ is a bounded sequence then the inequality holds for every $$0<p<1$$.

### MSC:

 42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

### Keywords:

Vilenkin-Fourier series; Hardy spaces; atomic decomposition
Full Text:

### References:

 [1] Gát, Gy., Investigations of certain operators with respect to the Vilenkin system, Acta Math. Hungar., 61, 131-149 (1993) · Zbl 0805.42019 [2] Schipp, F.; Wade, W. R.; Simon, P.; Pál, J., Walsh Series: An Introduction to Dyadic Harmonic Analysis (1990), Budapest Adam HilgerAkadémiai Kiadó: Budapest Adam HilgerAkadémiai Kiadó Bristol/New York · Zbl 0727.42017 [3] Simon, P., Investigations with respect to the Vilenkin system, Ann. Univ. Sci. Budapest Sect. Math., 28, 87-101 (1985) · Zbl 0586.43001 [4] Simon, P., Strong convergence of certain means with respect to the Walsh-Fourier series, Acta Math. Hungar., 49, 425-431 (1987) · Zbl 0643.42020 [5] P. Simon, Remarks on strong convergence with respect to the Walsh system, to appear.; P. Simon, Remarks on strong convergence with respect to the Walsh system, to appear. · Zbl 1084.42513 [6] Simon, P.; Weisz, F., Hardy-Littlewood type inequalities for Vilenkin-Fourier coefficients, Anal. Math., 24, 131-150 (1998) · Zbl 0913.42020 [7] Smith, B., A strong convergence theorem for $$H^1(T)$$, Lecture Notes in Math. (1983), Springer: Springer Berlin/New York, p. 169-173 [8] Vilenkin, N. Ja., On a class of complete orthogonormal system, Izv. Akad. Nauk. SSSR Ser. Mat., 11, 363-400 (1947) · Zbl 0036.35601 [9] Weisz, F., Martingale Hardy Spaces and their applications in Fourier analysis, Lecture Notes in Math. (1994), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0796.60049 [10] Weisz, F., Strong convergence theorem for two-parameter Walsh-Fourier and trigonometric-Fourier series, Studia Math., 117, 173-194 (1996) · Zbl 0839.42009 [11] Zygmund, A., Trigonometric Series (1959), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · JFM 58.0280.01
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