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TWO-SIDED ESTIMATES OF THE LEBESGUE CONSTANTS WITH RESPECT TO VILENKIN SYSTEMS AND APPLICATIONS

  • I. BLAHOTA (a1), L. E. PERSSON (a2) and G. TEPHNADZE (a3)

Abstract

In this paper, we derive two-sided estimates of the Lebesgue constants for bounded Vilenkin systems, we also present some applications of importance, e.g., we obtain a characterization for the boundedness of a subsequence of partial sums with respect to Vilenkin–Fourier series of H 1 martingales in terms of n's variation. The conditions given in this paper are in a sense necessary and sufficient.

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1. Agaev, G. N., Vilenkin, N. Ya., Dzhafarly, G. M. and Rubinstein, A. I., Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, Baku, Ehim, 1981 (in Russian).
2. Golubov, B. I., Efimov, A. V. and Skvortsov, V. A., Walsh series and transforms. (Russian) Nauka, Moscow, 1987, English transl. in Mathematics and its Applications (Soviet Series), vol. 64 (Kluwer Academic Publishers Group, Dordrecht, 1991).
3. Lukomskii, S. F., Lebesgue constants for characters of the compact zero-dimensional Abelian group, East. J. Appr. 15 (2) (2010), 219231.
4. Lukyanenko, O. A., On Lebesgue constants for Vilenkin system, in Mathematics. Mechanics (Saratov State University, 2005), no. 7, 7073 (in Russian).
5. Neveu, J., Discrete-parameter martingales, North-Holland Mathematical Library, vol. 10 (North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975).
6. Onneweer, C.W., On L-convergence of Walsh-Fourier series, Internat. J. Math. Sci. 1 (1978), 4756.
7. Schipp, F., Wade, W. R., Simon, P. and Pál, J., Walsh series. An Introduction to Dyadic Harmonic Analysis (Adam Hilger, Bristol-New York, 1990).
8. Tephnadze, G., On the partial sums of Vilenkin-Fourier series, J. Contemp. Math. Anal. 49 (1) (2014), 2332.
9. Tephnadze, G., Martingale Hardy spaces and summability of the one dimensional Vilenkin-Fourier series, PhD Thesis (Department of Engineering Sciences and Mathematics, Luleå University of Technology, Oct. 2015) (ISSN 1402-1544).
10. Vilenkin, N. Ya., A class of complete orthonormal systems, Izv. Akad. Nauk. U.S.S.R., Ser. Mat. 11 (1947), 363400.
11. Watari, C., Best approximation by Walsh polynomials, Tohoku Math. J. 15 (1963), 15.
12. Weisz, F., Martingale Hardy spaces and their applications in Fourier Analysis (Springer, Berlin-Heidelberg-New York, 1994).
13. Weisz, F., Hardy spaces and Cesàro means of two-dimensional Fourier series, Bolyai Soc. Math. Studies 5 (1996), 353367.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
  • URL: /core/journals/glasgow-mathematical-journal
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