Abstract and Applied Analysis

Volume 2016 (2016), Article ID 9712878, 10 pages

http://dx.doi.org/10.1155/2016/9712878

## On Estimates of Deviation of Functions from Matrix Operators of Their Fourier Series by Some Expressions with -Differences of the Entries

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Ulica Szafrana 4a, 65-516 Zielona Góra, Poland

Received 15 October 2015; Revised 11 February 2016; Accepted 23 March 2016

Academic Editor: Sergey Tikhonov

Copyright © 2016 Włodzimierz Łenski and Bogdan Szal. 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.

#### Abstract

We generalize the results of Krasniqi 2012 and Wei and Yu 2012 to the case of -differences.

#### 1. Introduction

Let or , where is the class of all –periodic real-valued functions, integrable in the Lebesgue sense with th power when and essentially bounded when [continuous], over with the norm and consider the trigonometric Fourier series with the partial sums .

Let be an infinite matrix of real numbers such that The -transformation of be given by In this paper, we study the upper bounds of by the second modulus of continuity of in the space defined by the formula where

The deviation with lower triangular infinite matrix such thatwas estimated in the sup norm by Krasniqi [1, Theorem , p. ] (see also [2]) as follows.

Theorem 1. *Let and satisfy (8). Then, where denotes the modulus of continuity of and if is such that where , we have Additionally, if satisfies the condition then *

In our theorems we will consider the -differences instead of considered above. We will formulate the general relation for like it formulated only for in [1, Theorem , p. ].

#### 2. Statement of the Results

Let us consider a function of modulus of continuity type on the interval , that is, a nondecreasing continuous function having the following properties: , for any .

Suppose that satisfies the conditionwhere is such that

Let Our main results on the degrees of approximation are the following.

Theorem 2. *If , where satisfies condition (14) such that (15) holds and , then Additionally, if a matrix is such thatthen *

Theorem 3. *If , where satisfies condition (14) such that (15) holds and , then *

Theorem 4. *Let , where satisfies condition (14) such that (15) holds and . If a matrix is such thatthen *

Theorem 5. *If and a matrix is such that (21) holds and , then *

Corollary 6. *For a lower triangular infinite matrix conditions (18) and (21) hold always and therefore for a lower triangular infinite matrix such that conditions (7) and (8) hold one can obtain the results from Theorems 2, 4, and 5 without the assumptions (18) and (21), where the mentioned results of Krasniqi follow as the special cases with Moreover, one can consider the essentially wider class of sequences than in the mentioned paper with the same degrees of approximation (see, e.g., [3, Theorem 2]).*

Corollary 7. *From Theorem 5 it follows that if where satisfies condition (14) such that (15) holds and a matrix is such that (21) andwith some and , hold, then *

*Remark 8. *The class of sequences defined by condition (24) was introduced by the second author in [4]. The similar classes were considered by Dyachenko and Tikhonov in [5] with (see also [6]).

*Remark 9. *We note that instead of (14) and (15) one can use Bari-Stechkin conditions Then all results are true for instead of and Lemmas 12 and 14 are not necessary.

#### 3. Auxiliary Results

We begin this section by some notations from [7]. Let for

It is clear by [8] that Hence

Next, we present the known estimates.

Lemma 10 (see [8]). *If then and, for any real , we have *

Lemma 11 (see [7]). *Let , , and . If then for every *

Lemma 12 (see [9]). *If (14) and (15) hold then *

We additionally proved two slight changed estimates.

Lemma 13. *If (14) and (15) hold, then, for and ,*

*Proof. *By substitution of and monotonicity of (see Lemma 15) we obtain, for , and the desired result follows from the Lemma 12.

Lemma 14. *If (14) and (15) hold, then, for ,*

*Proof. *Using Lemma 13 with , and (14) our result follows for

Finally, we present very useful property of the modulus of continuity.

Lemma 15 (see [8]). *A function of modulus of continuity type on the interval satisfies the following condition: *

#### 4. Proofs of the Results

*Proof of Theorem 2. *It is clear that for an odd and for an even Then, Since, by Lemma 11, thenHence, by Lemma 10, Using Lemmas 13 and 14 with and the estimates for and for , where , we obtain Analogously Similarly, by Lemmas 10, 13, and 14 with and the estimates for and for , where , we get Thus Collecting these estimates we obtain the first result.

Applying condition (18) we have and the second result also follows.

*Proof of Theorem 3. *Analogously, as in the proof of Theorem 2, we consider an odd and an even Then, or respectively, and Since Therefore, in the terms , , and we can estimate analogously as in the proof of Theorem 2 and thus we obtain the desired estimate.

*Proof of Theorem 4. *Similarly, as in the proof of Theorem 2, we consider an odd and an even Then, or respectively. Therefore, By Lemma 10 and (18), Further, by the same lemmas and conditions as above and Lemma 15, we obtain, with that Thus our proof is complete.

*Proof of Theorem 5. *Let as above Further, taking and , using Lemma 15, we obtain, with that