Generalized absolute convergence of single and double Vilenkin-Fourier series and related results

Nayna Govindbhai Kalsariya; Bhikha Lila Ghodadra

Mathematica Bohemica (2024)

  • Volume: 149, Issue: 2, page 129-166
  • ISSN: 0862-7959

Abstract

top
We consider the Vilenkin orthonormal system on a Vilenkin group G and the Vilenkin-Fourier coefficients f ^ ( n ) , n ∈ ℕ , of functions f ∈ L p ( G ) for some 1 < p ≤ 2 . We obtain certain sufficient conditions for the finiteness of the series ∑ n = 1 ∞ a n | 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).

How to cite

top

Kalsariya, Nayna Govindbhai, and Ghodadra, Bhikha Lila. "Generalized absolute convergence of single and double Vilenkin-Fourier series and related results." Mathematica Bohemica 149.2 (2024): 129-166. <http://eudml.org/doc/299419>.

@article{Kalsariya2024,
abstract = {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 $\lbrace a_n\rbrace $ 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).},
author = {Kalsariya, Nayna Govindbhai, Ghodadra, Bhikha Lila},
journal = {Mathematica Bohemica},
keywords = {generalized absolute convergence; Vilenkin-Fourier series; modulus of continuity; multiplicative system},
language = {eng},
number = {2},
pages = {129-166},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Generalized absolute convergence of single and double Vilenkin-Fourier series and related results},
url = {http://eudml.org/doc/299419},
volume = {149},
year = {2024},
}

TY - JOUR
AU - Kalsariya, Nayna Govindbhai
AU - Ghodadra, Bhikha Lila
TI - Generalized absolute convergence of single and double Vilenkin-Fourier series and related results
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 149
IS - 2
SP - 129
EP - 166
AB - 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 $\lbrace a_n\rbrace $ 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).
LA - eng
KW - generalized absolute convergence; Vilenkin-Fourier series; modulus of continuity; multiplicative system
UR - http://eudml.org/doc/299419
ER -

References

top
  1. Fine, N. J., 10.1090/S0002-9947-1949-0032833-2, Trans. Am. Math. Soc. 65 (1949), 372-414. (1949) Zbl0036.03604MR0032833DOI10.1090/S0002-9947-1949-0032833-2
  2. Folland, G. B., 10.1201/b19172, Textbooks in Mathematics. CRC Press, Boca Raton (2016). (2016) Zbl1342.43001MR3444405DOI10.1201/b19172
  3. Ghodadra, B. L., 10.5937/KgJMath1601091G, Kragujevac J. Math. 40 (2016), 91-104. (2016) Zbl1474.42108MR3509605DOI10.5937/KgJMath1601091G
  4. Gogoladze, L., Meskhia, R., On the absolute convergence of trigonometric Fourier series, Proc. A. Razmadze Math. Inst. 141 (2006), 29-40. (2006) Zbl1113.42004MR2259020
  5. Golubov, B., Efimov, A., Skvortsov, V., 10.1007/978-94-011-3288-6, Mathematics and Its Applications. Soviet Series 64. Kluwer, Dordrecht (1991). (1991) Zbl0785.42010MR1155844DOI10.1007/978-94-011-3288-6
  6. Golubov, B. I., Volosivets, S. S., 10.1007/s10476-012-0202-8, Anal. Math. 38 (2012), 105-122. (2012) Zbl1265.42005MR2925159DOI10.1007/s10476-012-0202-8
  7. Hewitt, E., Ross, K. A., 10.1007/978-1-4419-8638-2, Die Grundlehren der mathematischen Wissenschaften 115. Springer, Berlin (1963). (1963) Zbl0115.10603MR0156915DOI10.1007/978-1-4419-8638-2
  8. Izumi, M., Izumi, S., 10.1007/BF02591034, Ark. Mat. 7 (1967), 177-184. (1967) Zbl0189.07102MR0221195DOI10.1007/BF02591034
  9. Móricz, F., 10.1007/s10476-010-0402-z, Anal. Math. 36 (2010), 275-286. (2010) Zbl1240.42131MR2738321DOI10.1007/s10476-010-0402-z
  10. Móricz, F., Veres, A., 10.1007/s10474-010-0065-z, Acta Math. Hung. 131 (2011), 122-137. (2011) Zbl1240.42132MR2776656DOI10.1007/s10474-010-0065-z
  11. Onneweer, C. W., 10.1215/S0012-7094-72-03965-8, Duke Math. J. 39 (1972), 599-609. (1972) Zbl0252.43016MR0316976DOI10.1215/S0012-7094-72-03965-8
  12. Quek, T. S., Yap, L. Y. H., 10.1016/0022-247X(80)90110-9, J. Math. Anal. Appl. 74 (1980), 1-14. (1980) Zbl0434.43008MR0568369DOI10.1016/0022-247X(80)90110-9
  13. Volosivets, S. S., Kuznetsova, M. A., 10.1134/S0001434620010216, Math. Notes 107 (2020), 217-230. (2020) Zbl1442.42016MR4070005DOI10.1134/S0001434620010216
  14. Walker, P. L., 10.1017/S0305004100041906, Proc. Camb. Philos. Soc. 63 (1967), 923-928. (1967) Zbl0184.36301MR0216246DOI10.1017/S0305004100041906
  15. Younis, M. S., 10.1016/0022-247X(92)90273-G, J. Math. Anal. Appl. 163 (1992), 15-19. (1992) Zbl0752.43007MR1144701DOI10.1016/0022-247X(92)90273-G

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.