© Bogdan Szal. 2010
Received: 26 January 2010
Accepted: 13 April 2010
Published: 23 May 2010
We introduce a new class of sequences called and give a sufficient and necessary condition for weighted integrability of trigonometric series with coefficients to belong to the above class. This is a generalization of the result proved by M. Dyachenko and S. Tikhonov (2009). Then we discuss the relations among the weighted best approximation and the coefficients of trigonometric series. Moreover, we extend the results of B. Wei and D. Yu (2009) to the class .
In Subsection 2.1 we generalize the following result.
In Subsection 2.2 we generalize and extend the following results .
In order to formulate our new results we define the next class of sequences.
2. Statement of the Results
We formulate our results as follows.
If we take (and in Theorem 2.2), then the result of Dyachenko and Tikhonov [15, Theorems and ] follows from Theorems 2.2 and 2.3. By the embedding relations (1.8) and (1.15) we can also derive from Theorem 2.2 the result of You, Zhou, and Zhou .
2.2. Relations between The Best Approximation and Fourier Coefficients
If we restrict our attention to the class then by (1.8) and (1.15) Wei and Yu's result  follows from Theorems 2.5 and 2.6.
3. Auxiliary Results
4. Proofs of The Main Results
4.1. Proof of Theorem 2.1
4.2. Proof of Theorem 2.2
which completes the proof.
4.3. Proof of Theorem 2.3
This ends our proof.
4.4. Proof of Theorem 2.5
Combining (4.32) or (4.33), (4.35)–(4.43) and (4.45)–(4.50) we complete the proof of Theorem 2.5.
4.5. Proof of Theorem 2.6
This completes the proof.
- Heywood P: On the integrability of functions defined by trigonometric series. The Quarterly Journal of Mathematics 1954, 5: 71–76. 10.1093/qmath/5.1.71MathSciNetView ArticleMATHGoogle Scholar
- Chen Y-M: On the integrability of functions defined by trigonometrical series. Mathematische Zeitschrift 1956, 66: 9–12. 10.1007/BF01186590MathSciNetView ArticleMATHGoogle Scholar
- Zygmund A: Trigonometric Series. Cambridge University Press, Cambridge, Mass, USA; 1977.MATHGoogle Scholar
- Boas RP Jr.: Integrability Theorems for Trigonometric Transforms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 38. Springer, New York, NY, USA; 1967:v+66.Google Scholar
- Askey R, Wainger S: Integrability theorems for Fourier series. Duke Mathematical Journal 1966, 33: 223–228. 10.1215/S0012-7094-66-03326-6MathSciNetView ArticleMATHGoogle Scholar
- Leindler L: A new class of numerical sequences and its applications to sine and cosine series. Analysis Mathematica 2002, 28(4):279–286. 10.1023/A:1021717700626MathSciNetView ArticleMATHGoogle Scholar
- Tikhonov SYu: On the integrability of trigonometric series. Matematicheskie Zametki 2005, 78(3–4):437–442.MathSciNetMATHGoogle Scholar
- Tikhonov S: Trigonometric series with general monotone coefficients. Journal of Mathematical Analysis and Applications 2007, 326(1):721–735. 10.1016/j.jmaa.2006.02.053MathSciNetView ArticleMATHGoogle Scholar
- Yu DS, Zhou P, Zhou SP: On integrability and convergence of trigonometric series. Studia Mathematica 2007, 182(3):215–226. 10.4064/sm182-3-3MathSciNetView ArticleMATHGoogle Scholar
- Tikhonov S: Trigonometric series of Nikol'skii classes. Acta Mathematica Hungarica 2007, 114(1–2):61–78. 10.1007/s10474-006-0513-yMathSciNetView ArticleMATHGoogle Scholar
- Tikhonov S: Best approximation and moduli of smoothness: computation and equivalence theorems. Journal of Approximation Theory 2008, 153(1):19–39. 10.1016/j.jat.2007.05.006MathSciNetView ArticleMATHGoogle Scholar
- Le RJ, Zhou SP: A new condition for the uniform convergence of certain trigonometric series. Acta Mathematica Hungarica 2005, 108(1–2):161–169. 10.1007/s10474-005-0217-8MathSciNetView ArticleMATHGoogle Scholar
- Leindler L: Embedding results regarding strong approximation of sine series. Acta Scientiarum Mathematicarum 2005, 71(1–2):91–103.MathSciNetMATHGoogle Scholar
- You DS, Zhou P, Zhou SP: Ultimate generalization to monotonicity for uniform convergence of trigonometric series. preprint, http://arxiv.org/abs/math/0611805
- Dyachenko M, Tikhonov S: Integrability and continuity of functions represented by trigonometric series: coefficients criteria. Studia Mathematica 2009, 193(3):285–306. 10.4064/sm193-3-5MathSciNetView ArticleMATHGoogle Scholar
- Wei B, Yu D: On weighted integrability and approximation of trigonometric series. Analysis in Theory and Applications 2009, 25(1):40–54. 10.1007/s10496-009-0040-0MathSciNetView ArticleMATHGoogle Scholar
- Konyushkov AA: Best approximation by trigonometric polynomials and Fourier coefficients. Sbornik Mathematics 1958, 44: 53–84.Google Scholar
- Leindler L: Best approximation and Fourier coefficients. Analysis Mathematica 2005, 31(2):117–129. 10.1007/s10476-005-0008-zMathSciNetView ArticleMATHGoogle Scholar
- Szal B: On -convergence of trigonometric series. to appear, http://arxiv.org/abs/0908.4578
- Leindler L: Generalization of inequalities of Hardy and Littlewood. Acta Scientiarum Mathematicarum 1970, 31: 279–285.MathSciNetMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.