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# On Weighted Integrability of Functions Defined by Trigonometric Series

*Journal of Inequalities and Applications*
**volume 2010**, Article number: 485705 (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 .

## 1. Introduction

Let , , be the space of all -power integrable functions of period equipped with the norm

Write

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

For and a sequence let

In Subsection 2.1 we generalize the following result.

Theorem 1.1.

Let a nonnegative sequence , and . Then

In the case when denotes the class of all decreasing sequences, this theorem was proved in [1–4]; for , the class of quasimonotone sequences, in [5]; for in [6, 7]; for in [8]; and for in [9], where

In [15] Dyachenko and Tikhonov extended Theorem 1.1 to the class , where and

We have (see [15])

Let be a nonnegative function defined on the interval . Denote by the best approximation of by trigonometric polynomials of degree at most in the weighted -norm, that is,

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

A sequence of nonnegative terms is called almost increasing (decreasing) if there exists a constant such that

We say that a weight function if is defined by the sequence as follows: , , and there exist positive constants and such that

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.

Assume that . If for some and , then for

Theorem 1.3.

Assume that . If for some and , then for

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.

Let and . One says that a sequence belongs to , if the relation

*holds for all*.

Note that for and (see Theorem 2.1(i))

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

(ii)Let , and . If , then

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

### 2.1. Weighted -Integrability

Let and . We define on the interval an even function , which is given on the interval by the formula

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

Theorem 2.2.

Let a nonnegative sequence , where , and . If

then if and only if

Theorem 2.3.

Let a nonnegative sequence , , and . If

then if and only if

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.

Let a nonnegative sequence , where , , and . If

and (2.3) holds, then

where .

Theorem 2.6.

Let a nonnegative sequence , and . If

and (2.5) holds, then

where .

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.

## 4. Proofs of The Main Results

### 4.1. Proof of Theorem 2.1

(i)Let , and

First, we prove that . Let

Then for all

and . If then

and since

the inequality

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

whence . Thus .

Let and and let

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.

*Sufficiency.* Suppose that (2.3) holds. Then

It is clear that for an odd

(for the last sum should be omitted), and for an even

First, we estimate the following integral:

By (3.3), for we have

Using (3.2) with and the inequality

we get

If then by (3.3), for we obtain

Now, we estimate the following integral:

By (3.3), for we have

Using (3.2) with and the inequality

we obtain

If then by (3.3), for , we obtain

Thus, combining (4.9), (4.12)–(4.13), (4.16)–(4.18), (4.21), and (4.10) or (4.11), we obtain that

Necessity.

We follow the method adopted by . Tikhonov [15]. Note that if , then . Integrating , we have

and consequently

If , then using (4.24),

Using this and (3.3), for , we obtain

Defining we get

and by (3.3), for , we obtain

Applying Hölder's inequality, for we have

Finally,

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.

Applying the inequalities for , for and using (3.3), for we have

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.

If then by (2.3) we obtain that (2.7) obviously holds. Let . It is clear that if is an odd number, then

(for the last sum should be omitted), and if is an even number, then

Let

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

Then, for , by (3.2) and (4.14), we get

We immediately have for

If and , then by Hölder's inequality we have for

When and , an elementary calculation gives for

If , then

We have

and taking and for , for in (3.3), we get for

If , then using (3.2) and (4.14), we have

Set , for and for . Then by (3.3), we have for

Let

where and . Then, for , using (3.2) and (4.19), we get

Therefore,

If , then

Similarly as in the estimation of the quantity using (3.3) for , we have

If , then using (3.2) and (4.19), we have

Further, by (3.3), we have for

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.

Applying the inequalities for and for and using (3.3), for we have

This completes the proof.

## References

Heywood P: On the integrability of functions defined by trigonometric series.

*The Quarterly Journal of Mathematics*1954, 5: 71–76. 10.1093/qmath/5.1.71Chen Y-M: On the integrability of functions defined by trigonometrical series.

*Mathematische Zeitschrift*1956, 66: 9–12. 10.1007/BF01186590Zygmund A:

*Trigonometric Series*. Cambridge University Press, Cambridge, Mass, USA; 1977.Boas RP Jr.:

*Integrability Theorems for Trigonometric Transforms, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 38*. Springer, New York, NY, USA; 1967:v+66.Askey R, Wainger S: Integrability theorems for Fourier series.

*Duke Mathematical Journal*1966, 33: 223–228. 10.1215/S0012-7094-66-03326-6Leindler 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:1021717700626Tikhonov SYu: On the integrability of trigonometric series.

*Matematicheskie Zametki*2005, 78(3–4):437–442.Tikhonov S: Trigonometric series with general monotone coefficients.

*Journal of Mathematical Analysis and Applications*2007, 326(1):721–735. 10.1016/j.jmaa.2006.02.053Yu DS, Zhou P, Zhou SP: On integrability and convergence of trigonometric series.

*Studia Mathematica*2007, 182(3):215–226. 10.4064/sm182-3-3Tikhonov S: Trigonometric series of Nikol'skii classes.

*Acta Mathematica Hungarica*2007, 114(1–2):61–78. 10.1007/s10474-006-0513-yTikhonov S: Best approximation and moduli of smoothness: computation and equivalence theorems.

*Journal of Approximation Theory*2008, 153(1):19–39. 10.1016/j.jat.2007.05.006Le 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/s10474-005-0217-8Leindler L: Embedding results regarding strong approximation of sine series.

*Acta Scientiarum Mathematicarum*2005, 71(1–2):91–103.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/sm193-3-5Wei B, Yu D: On weighted integrability and approximation of trigonometric series.

*Analysis in Theory and Applications*2009, 25(1):40–54. 10.1007/s10496-009-0040-0Konyushkov AA: Best approximation by trigonometric polynomials and Fourier coefficients.

*Sbornik Mathematics*1958, 44: 53–84.Leindler L: Best approximation and Fourier coefficients.

*Analysis Mathematica*2005, 31(2):117–129. 10.1007/s10476-005-0008-zSzal 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.

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Szal, B. On Weighted Integrability of Functions Defined by Trigonometric Series.
*J Inequal Appl* **2010, **485705 (2010). https://doi.org/10.1155/2010/485705

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

### Keywords

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