- Research Article
- Open Access

- Bogdan Szal
^{1}Email author

**2010**:485705

https://doi.org/10.1155/2010/485705

© Bogdan Szal. 2010

**Received:**26 January 2010**Accepted:**13 April 2010**Published:**23 May 2010

## Abstract

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 .

## Keywords

- Positive Constant
- Weight Function
- Integrable Function
- Fourier Coefficient
- Elementary Calculation

## 1. Introduction

for those where the series converge. Denote by either or , and let be its associated coefficients, that is, is either or .

In Subsection 2.1 we generalize the following result.

Theorem 1.1.

where denotes the set of all trigonometric polynomials of degree at most .

for all , and the sequences , are almost decreasing and almost increasing, respectively.

In Subsection 2.2 we generalize and extend the following results [16].

Theorem 1.2.

Theorem 1.3.

If and or the above theorem has been obtained by Konyushkov [17] and Leindler [18] for , respectively.

In order to formulate our new results we define the next class of sequences.

Definition 1.4.

Throughout this paper, we use to denote a positive constant independent of the integer ; may depend on the parameters such as and , and it may have different values in different occurrences.

## 2. Statement of the Results

We formulate our results as follows.

Theorem 2.1.

Suppose that . The following properties are true.

(i)For any , and there exists a sequence , which does not belong to the class

(iii)Let and . If and , then the classes and are not comparable.

### 2.1. Weighted -Integrability

where if is an odd number, and if is an even number; for , and for .

Theorem 2.2.

Theorem 2.3.

Remark 2.4.

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 [9].

### 2.2. Relations between The Best Approximation and Fourier Coefficients

Theorem 2.5.

Theorem 2.6.

Remark 2.7.

If we restrict our attention to the class then by (1.8) and (1.15) Wei and Yu's result [16] follows from Theorems 2.5 and 2.6.

## 3. Auxiliary Results

## 4. Proofs of The Main Results

### 4.1. Proof of Theorem 2.1

does not hold, that is, does not belong to .

Let , and . If , then exists a natural number such that Supposing that , we have for all

Supposing that and , we can prove, similarly as in , that , , and Therefore the classes and are not comparable.

### 4.2. Proof of Theorem 2.2

We prove the theorem for the case when . The case when can be proved similarly.

Necessity.

which completes the proof.

### 4.3. Proof of Theorem 2.3

The proof of Theorem 2.3 goes analogously as the proof of Theorem 2.2. The only difference is that instead of (4.13) (for ) and (4.18) (for ) we use the below estimations.

This ends our proof.

### 4.4. Proof of Theorem 2.5

We prove the theorem for the case when . The case when can be proved similarly.

where and if is an even number, and if is an odd number.

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

The proof of Theorem 2.6 goes analogously as the proof of Theorem 2.5. The only difference is that instead of (4.41) (for ) and (4.48) (for ) we use the below estimations.

This completes the proof.

## Authors’ Affiliations

## References

- 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

## Copyright

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.