| Title:
|
Generalized absolute convergence of single and double Vilenkin-Fourier series and related results (English) |
| Author:
|
Kalsariya, Nayna Govindbhai |
| Author:
|
Ghodadra, Bhikha Lila |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
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149 |
| Issue:
|
2 |
| Year:
|
2024 |
| Pages:
|
129-166 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider the Vilenkin orthonormal system on a Vilenkin group $G$ and the Vilenkin-Fourier coefficients $\hat {f}(n)$, $n\in \mathbb {N}$, of functions $f\in L^p(G)$ for some $1<p\le 2$. We obtain certain sufficient conditions for the finiteness of the series $\sum _{n=1}^{\infty }a_n|\hat {f}(n)|^r$, where $\{a_n\}$ is a given sequence of positive real numbers satisfying a mild assumption and $0<r<2$. We also find analogous conditions for the double Vilenkin-Fourier series. These sufficient conditions are in terms of (either global or local) moduli of continuity of $f$ and give multiplicative analogue of some results due to Móricz (2010), Móricz and Veres (2011), Golubov and Volosivets (2012), and Volosivets and Kuznetsova (2020). (English) |
| Keyword:
|
generalized absolute convergence |
| Keyword:
|
Vilenkin-Fourier series |
| Keyword:
|
modulus of continuity |
| Keyword:
|
multiplicative system |
| MSC:
|
42C10 |
| DOI:
|
10.21136/MB.2023.0023-22 |
| . |
| Date available:
|
2024-07-10T15:01:44Z |
| Last updated:
|
2024-07-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152464 |
| . |
| Reference:
|
[1] Fine, N. J.: On the Walsh functions.Trans. Am. Math. Soc. 65 (1949), 372-414. Zbl 0036.03604, MR 0032833, 10.1090/S0002-9947-1949-0032833-2 |
| Reference:
|
[2] Folland, G. B.: A Course in Abstract Harmonic Analysis.Textbooks in Mathematics. CRC Press, Boca Raton (2016). Zbl 1342.43001, MR 3444405, 10.1201/b19172 |
| Reference:
|
[3] Ghodadra, B. L.: On $\beta$-absolute convergence of Vilenkin-Fourier series with small gaps.Kragujevac J. Math. 40 (2016), 91-104. Zbl 1474.42108, MR 3509605, 10.5937/KgJMath1601091G |
| Reference:
|
[4] Gogoladze, L., Meskhia, R.: On the absolute convergence of trigonometric Fourier series.Proc. A. Razmadze Math. Inst. 141 (2006), 29-40. Zbl 1113.42004, MR 2259020 |
| Reference:
|
[5] Golubov, B., Efimov, A., Skvortsov, V.: Walsh Series and Transforms: Theory and Applications.Mathematics and Its Applications. Soviet Series 64. Kluwer, Dordrecht (1991). Zbl 0785.42010, MR 1155844, 10.1007/978-94-011-3288-6 |
| Reference:
|
[6] Golubov, B. I., Volosivets, S. S.: Generalized absolute convergence of single and double Fourier series with respect to multiplicative systems.Anal. Math. 38 (2012), 105-122. Zbl 1265.42005, MR 2925159, 10.1007/s10476-012-0202-8 |
| Reference:
|
[7] Hewitt, E., Ross, K. A.: Abstract Harmonic Analysis. Vol. I. Structure of Topological Groups. Integration Theory. Group Representations.Die Grundlehren der mathematischen Wissenschaften 115. Springer, Berlin (1963). Zbl 0115.10603, MR 0156915, 10.1007/978-1-4419-8638-2 |
| Reference:
|
[8] Izumi, M., Izumi, S.: On absolute convergence of Fourier series.Ark. Mat. 7 (1967), 177-184. Zbl 0189.07102, MR 0221195, 10.1007/BF02591034 |
| Reference:
|
[9] Móricz, F.: Absolute convergence of Walsh-Fourier series and related results.Anal. Math. 36 (2010), 275-286. Zbl 1240.42131, MR 2738321, 10.1007/s10476-010-0402-z |
| Reference:
|
[10] Móricz, F., Veres, A.: Absolute convergence of double Walsh-Fourier series and related results.Acta Math. Hung. 131 (2011), 122-137. Zbl 1240.42132, MR 2776656, 10.1007/s10474-010-0065-z |
| Reference:
|
[11] Onneweer, C. W.: Absolute convergence of Fourier series on certain groups.Duke Math. J. 39 (1972), 599-609. Zbl 0252.43016, MR 0316976, 10.1215/S0012-7094-72-03965-8 |
| Reference:
|
[12] Quek, T. S., Yap, L. Y. H.: Absolute convergence of Vilenkin-Fourier series.J. Math. Anal. Appl. 74 (1980), 1-14. Zbl 0434.43008, MR 0568369, 10.1016/0022-247X(80)90110-9 |
| Reference:
|
[13] Volosivets, S. S., Kuznetsova, M. A.: Generalized absolute convergence of single and double series in multiplicative systems.Math. Notes 107 (2020), 217-230. Zbl 1442.42016, MR 4070005, 10.1134/S0001434620010216 |
| Reference:
|
[14] Walker, P. L.: Lipschitz classes on 0-dimensional groups.Proc. Camb. Philos. Soc. 63 (1967), 923-928. Zbl 0184.36301, MR 0216246, 10.1017/S0305004100041906 |
| Reference:
|
[15] Younis, M. S.: On the absolute convergence of Vilenkin-Fourier series.J. Math. Anal. Appl. 163 (1992), 15-19. Zbl 0752.43007, MR 1144701, 10.1016/0022-247X(92)90273-G |
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