Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-10T16:52:17.670Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  13 March 2017

I. BLAHOTA ,
L. E. PERSSON  and
G. TEPHNADZE
I. BLAHOTA
Affiliation:
Institute of Mathematics and Computer Sciences, University of Nyíregyháza, P.O. Box 166, Nyíregyháza, H-4400, Hungary e-mail: blahota.istvan@nye.hu
L. E. PERSSON
Affiliation:
Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden UiT, The Artic University of Norway, P.O. Box 385, N-8505, Narvik, Norway e-mail: larserik@ltu.se
G. TEPHNADZE
Affiliation:
Department of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University, Chavchavadze str. 1, Tbilisi 0128, Georgia Department of Engineering Sciences and Mathematics, Luleå University of Technology, SE-971 87 Luleå, Sweden e-mail: giorgitephnadze@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 H1 martingales in terms of n's variation. The conditions given in this paper are in a sense necessary and sufficient.

MSC classification

Primary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 60 , Issue 1 , January 2018 , pp. 17 - 34
Copyright
Copyright © Glasgow Mathematical Journal Trust 2017 

References

REFERENCES

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).Google Scholar
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).Google Scholar
3. Lukomskii, S. F., Lebesgue constants for characters of the compact zero-dimensional Abelian group, East. J. Appr. 15 (2) (2010), 219–231.Google Scholar
4. Lukyanenko, O. A., On Lebesgue constants for Vilenkin system, in Mathematics. Mechanics (Saratov State University, 2005), no. 7, 70–73 (in Russian).Google Scholar
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).Google Scholar
6. Onneweer, C.W., On L-convergence of Walsh-Fourier series, Internat. J. Math. Sci. 1 (1978), 47–56.Google Scholar
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).Google Scholar
8. Tephnadze, G., On the partial sums of Vilenkin-Fourier series, J. Contemp. Math. Anal. 49 (1) (2014), 23–32.Google Scholar
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).Google Scholar
10. Vilenkin, N. Ya., A class of complete orthonormal systems, Izv. Akad. Nauk. U.S.S.R., Ser. Mat. 11 (1947), 363–400.Google Scholar
11. Watari, C., Best approximation by Walsh polynomials, Tohoku Math. J. 15 (1963), 1–5.Google Scholar
12. Weisz, F., Martingale Hardy spaces and their applications in Fourier Analysis (Springer, Berlin-Heidelberg-New York, 1994).Google Scholar
13. Weisz, F., Hardy spaces and Cesàro means of two-dimensional Fourier series, Bolyai Soc. Math. Studies 5 (1996), 353–367.Google Scholar