#### Abstract

In this paper, we give a characterization of Nikol’skiĭ-Besov type classes of functions, given by integral representations of moduli of smoothness, in terms of series over the moduli of smoothness. Also, necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to such a class are given. In order to prove our results, we make use of certain recent reverse Copson-type and Leindler-type inequalities.

#### 1. Introduction

Let , , be a -periodic function. We say that the function has monotone Fourier coefficients if it has a cosine Fourier series with

We say that the function has lacunary Fourier coefficients ifwherethat is,

By we denote the modulus of smoothness of order in metrics of a function , : whereis the th-order shift operator.

By we denote the best approximation in metrics of a function , , by means of trigonometric polynomials whose degree is not greater than ; that is, where and and are arbitrary real numbers.

We say that a -periodic function belongs to the Nikol’skiĭ-Besov class , , if the following conditions are satisfied:(1).(2)Numbers , , and belong to the interval , and is an integer satisfying .(3)The following inequality holds true:while the function satisfies the following conditions:(4) is a nonnegative continuous function on and .(5)For every and , where , holds.(6)For every , where , holds.Constants (without mentioning it explicitly, we will consider all the constants positive) , , and do not depend on , , and .

A more detailed approach to the classes is given in [1, 2] (see also [3]). In the paper, we give a characterization of classes of functions in terms of series over their moduli of smoothness. Then we give the necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to a class . In the process of proving the results, we make use of certain recent reverse -type inequalities [4], closely related to Copson’s and Leindler’s inequalities.

Finally, by making use of our results, we construct an example of a function having a lacunary Fourier series, which shows that classes are properly embedded between the appropriate Nikol’skiĭ classes and Besov classes.

#### 2. Statement of Results

Now we formulate our results.

Theorem 1. *A function belongs to the class if and only if (here and below we assume that the parameters , , , and satisfy condition (2) and the function satisfies conditions (4)–(6) of the definition of the class )where constant does not depend on .*

Theorem 2. *For a function , , such thatto belong to the class , it is necessary and sufficient that its Fourier coefficients satisfy the conditionwhere constant does not depend on .*

Regarding Theorem 1, a very interesting open question on its analogue for functions with general monotone Fourier coefficients generalized in the sense of [5, 6] remains.

Corollary 3. *Putting , , in the definition of the class , we obtain [1] the Nikol’skiĭ class . Thus Theorems 1 and 2 give the single coefficient condition for , given in [7], where the function is given by (10).*

Corollary 4. *If , then we obtain [1] the Besov class . Thus Theorems 1 and 2 give the necessary and sufficient conditionfor , given in [8], where the function is given by (10).*

Theorem 5. *For a function , , such that to belong to the class , it is necessary and sufficient that its Fourier coefficients satisfy the condition (here and below we assume that the parameters , , , and satisfy condition (2) and the function satisfies conditions (4)–(6).)where constant does not depend on .*

Corollary 6. *Putting , , in the definition of the class , we obtain [1] the Nikol’skiĭ class . Thus Theorem 5 gives the single coefficient conditionfor , where the function is given by (14).*

Corollary 7. *If , then we obtain [1] the Besov class . Thus Theorem 5 gives the necessary and sufficient condition for , given in [8], where the function is given by (14).*

*Example 8. *LetwhereThen, we havethus implying (see the proof of Theorem 5) that for . This means that classes are classes of embedding between classes and .

#### 3. Auxiliary Statements

In order to establish our results, we use the following lemmas.

Lemma 9. *Let and . The following inequality holds true: *

Proof of the lemma is due to Jensen [9, p. 43].

Lemma 10. *Let be a sequence of nonnegative numbers, let , let be a real number, and let and be positive integers such that . Then*(1)*for , the following equalities hold: *(2)*for , the following equalities hold:**where constants , , , and depend only on numbers , , and and do not depend on and as well as on the sequence .*

Proof of the lemma is given in [9, p. 308].

Lemmas 11 and 12 state certain -type inequalities which are reversed to the ones given in Lemma 10 and closely related to Copson’s and Leindler’s inequalities (see, e.g., [10–13]).

We write if is a monotone-decreasing sequence of nonnegative numbers, that is, if .

Lemma 11. *Let , let , let be a real number, and let and be positive integers. Then*(1)*for and , the following equalities hold:*(2)*for and , the following equalities hold:**where constants , , , and depend only on numbers , , and and do not depend on and as well as on the sequence .*

Proof of the lemma is given in [4].

Lemma 12. *Let , let , let be a real number, and let and be positive integers. For , the following inequalities hold:where constants , , , and depend only on numbers , , and and do not depend on and as well as on the sequence .*

The lemma is also proven in [4].

Lemma 13. *Let for fixed from the interval and let The following inequalities hold:where constants and do not depend on and .*

The lemma is proven in [8].

Lemma 14. *A function belongs to the class if and only if where constant does not depend on .*

Proof of the lemma is given in [1].

Lemma 15. *Let , , and The following inequalities hold: where constants and do not depend on .*

Proof of the lemma is due to Zygmund [14, vol. I, p. 326].

Corollary 16. *Lemma 15 yields the following estimate:where constants and do not depend on and .*

#### 4. Proofs

Now we prove our results.

*Proof of Theorem 1. *PutWe have [9, p. 55]and, taking into account properties of modulus of smoothness [15], In an analogous way, we estimate Let . For a positive integer , we put . Then we have Hence, we obtainwhich proves inequality (9).

Now we suppose that inequality (9) holds. For , we choose the positive integer satisfying . Then, taking into consideration the estimates from above for and , we haveHenceimplying that .

Proof of Theorem 1 is completed.

*Proof of Theorem 2. *Theorem 1 implies that the condition is equivalent to the conditionwhere constant does not depend on . Lemma 13 yields that the last estimate is equivalent to the estimate where constant does not depend on . Hence, if we denote the terms on the left-hand side of the inequality by , , , and , respectively, then condition is equivalent to the condition Now we estimate the terms , , , and from below and above by means of expression taking part in the condition of the theorem.

First we estimate and from below. We haveFor , making use of Lemmas 10 and 11, we obtainIn an analogous way, for , we getWe estimate the term from above: For , we have and applying once more Lemmas 10 and 11, we obtainPutThen, fortaking into account the fact that and , we getSince , we have Applying Lemma 12, we obtainFrom (50), it follows that This way, inequalities (46), (47), (48), and (56) yield Now we estimate and . Put Applying Lemma 12, for , we get We estimate in an analogous way:We estimate the seriesFirst let . Applying Hölder’s inequality, we have Since , we get So, for , we have proven that Let . For given , we choose the positive integer such that . Then, we have Making use of Lemma 9, we obtain Since, for , holds, we getThis way, for , we proved thatHence, (60) yieldsNow, from (59), it follows thatFurther, we estimate the series whereHenceMaking use of (73) and (70), we haveHence, applying (73) in (57), we obtainNow we estimate and from below. Making use of Lemma 12, we get and, in an analogous way,Hence,This way the following inequality holds:From (57), it follows that Since holds, we haveNow, estimates (80) and (75) imply that This way we proved that condition (9) is equivalent to the condition of the theorem. Since condition (9) is equivalent to the condition , proof of Theorem 2 is completed.

*Proof of Theorem 5. *Considering Lemma 14, condition is equivalent to the condition where constant does not depend on . Corollary 16 yields that the last estimate is equivalent to the estimate where constant does not depend on .

Putwe estimate and from below and above.

Let . Using Lemma 9, changing the order of summation, we get Therefrom, taking into consideration the fact that while computing the second sum, we obtain Let and . Applying Hölder’s inequality, we have where . Computing the second sum, we obtain Now we haveThis way, for , we have where constant does not depend on .

Now we estimate from below.

Let . Making use of Lemma 9, we get Computing the second sum, we get Let and . Applying Hölder’s inequality, we have where . The last estimate implies that Changing the order of summation and then computing the second sum, we obtain where constant does not depend on .

Consequently, for every , the following estimate holds: where constants and do not depend on .

Now we estimate . Obviously, Let . Applying Lemma 9, changing the order of summation and then computing the second sum, we obtain Let and . Applying Hölder’s inequality, we get where . The last estimate implies that Changing the order of summation and computing the second sum, we have Thus, for every , the following holds: Now we estimate from above. Taking into consideration the fact that , we have Sinceholds and an upper bound for is already found, we estimate from above the expression Let . Applying Lemma 9, we obtain Let and . Then, applying Hölder’s inequality, we have where . Using the last estimate, we get Changing the order of summation and computing the second sum, we obtain Therefore, for every , the following estimate holds: Now, making use of inequalities (105) and (98), we have