On Weighted Integrability of Functions Defined by Trigonometric Series

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

(1.1)

Write

(1.2)

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

(1.3)

In Subsection 2.1 we generalize the following result.

Theorem 1.1.

Let a nonnegative sequence , and . Then

(1.4)

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

(1.5)

Note that (see [6, 8, 10–14])

(1.6)

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

(1.7)

We have (see [15])

(1.8)

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,

(1.9)

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

(1.10)

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

(1.11)

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

(1.12)

Theorem 1.3.

Assume that . If for some and , then for

(1.13)

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

(1.14)

holds for all.

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

(1.15)

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

(2.1)

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

(2.2)

then if and only if

(2.3)

Theorem 2.3.

Let a nonnegative sequence , , and . If

(2.4)

then if and only if

(2.5)

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

(2.6)

and (2.3) holds, then

(2.7)

where .

Theorem 2.6.

Let a nonnegative sequence , and . If

(2.8)

and (2.5) holds, then

(2.9)

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.

3. Auxiliary Results

Denote, for ,

(3.1)

Lemma 3.1 (see [19]).

Let , and . If , then for all

(3.2)

Lemma 3.2 (see [20]).

Let , and , then

(3.3)

4. Proofs of The Main Results

4.1. Proof of Theorem 2.1

(i)Let , and

(4.1)

First, we prove that . Let

(4.2)

Then for all

(4.3)

and . If then

(4.4)

and since

(4.5)

the inequality

(4.6)

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

(4.7)

whence . Thus .

Let and and let

(4.8)

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

(4.9)

It is clear that for an odd

(4.10)

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

(4.11)

First, we estimate the following integral:

(4.12)

By (3.3), for we have

(4.13)

Using (3.2) with and the inequality

(4.14)

we get

(4.15)

If then by (3.3), for we obtain

(4.16)

Now, we estimate the following integral:

(4.17)

By (3.3), for we have

(4.18)

Using (3.2) with and the inequality

(4.19)

we obtain

(4.20)

If then by (3.3), for , we obtain

(4.21)

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

(4.22)

Necessity.

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

(4.23)

and consequently

(4.24)

If , then using (4.24),

(4.25)

Using this and (3.3), for , we obtain

(4.26)

Defining we get

(4.27)

and by (3.3), for , we obtain

(4.28)

Applying Hölder's inequality, for we have

(4.29)

Finally,

(4.30)

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

(4.31)

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

(4.32)

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

(4.33)

Let

(4.34)

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

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

(4.35)

We immediately have for

(4.36)

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

(4.37)

When and , an elementary calculation gives for

(4.38)

If , then

(4.39)

We have

(4.40)

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

(4.41)

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

(4.42)

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

(4.43)

Let

(4.44)

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

(4.45)

Therefore,

(4.46)

If , then

(4.47)

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

(4.48)

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

(4.49)

Further, by (3.3), we have for

(4.50)

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

(4.51)

This completes the proof.