| Title:
|
Necessary conditions for the $L^{p}$-convergence $(0<p<1)$ of single and double trigonometric series (English) |
| Author:
|
Krasniqi, Xhevat Z. |
| Author:
|
Kórus, Péter |
| Author:
|
Móricz, Ferenc |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
139 |
| Issue:
|
1 |
| Year:
|
2014 |
| Pages:
|
75-88 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the $L^{p}$-metric, where $0<p<1$. The results and their proofs have been motivated by the recent papers of A. S. Belov (2008) and F. Móricz (2010). Our basic tools in the proofs are the Hardy-Littlewood inequality for functions in $H^{p}$ and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the $L^{p}$-metric, where $0<p<1$. (English) |
| Keyword:
|
trigonometric series |
| Keyword:
|
Hardy-Littlewood inequality for functions in $H^{p}$ |
| Keyword:
|
Bernstein-Zygmund inequalities for the derivative of trigonometric polynomials in $L^{p}$-metric for $0<p<1$ |
| Keyword:
|
necessary conditions for the convergence in $L^{p}$-metric |
| MSC:
|
42A16 |
| MSC:
|
42A20 |
| MSC:
|
42A32 |
| MSC:
|
42B05 |
| MSC:
|
42B30 |
| MSC:
|
42B99 |
| idZBL:
|
Zbl 06362243 |
| idMR:
|
MR3231430 |
| DOI:
|
10.21136/MB.2014.143637 |
| . |
| Date available:
|
2014-03-20T08:30:35Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143637 |
| . |
| Reference:
|
[1] Arestov, V.: On integral inequalities for trigonometric polynomials and their derivatives.Math. USSR, Izv. 18 (1982), 1-18 translation from Izv. Akad. Nauk SSSR, Ser. Mat. 45 (1981), 3-22. MR 0607574, 10.1070/IM1982v018n01ABEH001375 |
| Reference:
|
[2] Belov, A. S.: On conditions of the average convergence (upper boundedness) of trigonometric series.J. Math. Sci., New York 155 5-17 (2008), translation from Sovrem. Mat., Fundam. Napravl. 25 (2007), 8-20. MR 2342534, 10.1007/s10958-008-9204-2 |
| Reference:
|
[3] Bustamante, J.: Algebraic Approximation: A Guide to Past and Current Solutions.Frontiers in Mathematics Birkhäuser, Basel (2012). Zbl 1248.41012, MR 3014919 |
| Reference:
|
[4] Hardy, G. H., Littlewood, J. E.: Some new properties of Fourier constants.Math. Ann. 97 (1927), 159-209. MR 1512359, 10.1007/BF01447865 |
| Reference:
|
[5] Krasniqi, X. Z.: On the convergence (upper boundness) of trigonometric series.Math. Commun. 14 (2009), 245-254. Zbl 1204.42006, MR 2743173 |
| Reference:
|
[6] Móricz, F.: Necessary conditions for $L^1$-convergence of double Fourier series.J. Math. Anal. Appl. 363 (2010), 559-568. Zbl 1182.42009, MR 2564875, 10.1016/j.jmaa.2009.09.030 |
| Reference:
|
[7] Runovski, K., Schmeisser, H.-J.: On some extensions of Bernstein's inequalities for trigonometric polynomials.Funct. Approx. Comment. Math. 29 (2001), 125-142. MR 2135603, 10.7169/facm/1538186723 |
| . |
