On Estimates of Deviation of Functions from Matrix Operators of Their Fourier Series by Some Expressions with -Differences of the Entries
We generalize the results of Krasniqi 2012 and Wei and Yu 2012 to the case of -differences.
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
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 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]).
3. Auxiliary Results
We begin this section by some notations from . Let for
It is clear by  that Hence
Next, we present the known estimates.
Lemma 10 (see ). If then and, for any real , we have
Lemma 11 (see ). Let , , and . If then for every
We additionally proved two slight changed estimates.
Finally, we present very useful property of the modulus of continuity.
Lemma 15 (see ). 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.