Journal of Function Spaces
Volume 2015 (2015), Article ID 470205, 9 pages
http://dx.doi.org/10.1155/2015/470205

Research Article

On Pointwise Approximation of Conjugate Functions by Some Hump Matrix Means of Conjugate Fourier Series

W. Łenski and B. Szal

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

Received 20 October 2014; Revised 7 January 2015; Accepted 21 January 2015

Academic Editor: Rodolfo H. Torres

Copyright © 2015 W. Łenski and B. 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.

The results generalizing some theorems on summability are shown. The same degrees of pointwise approximation as in earlier papers by weaker assumptions on considered functions and examined summability methods are obtained. From presented pointwise results, the estimation on norm approximation is derived. Some special cases as corollaries are also formulated.

1. Introduction

Let (resp., ) be the class of all -periodic real-valued functions integrable in the Lebesgue sense with th power (essentially bounded) over with the norm

and consider the conjugate trigonometric Fourier series

with the partial sums . We know that if then

where

with

exists for almost all [1, Th.(3.1)IV].

Let and be infinite lower triangular matrices of real numbers such that

and let, for ,

Let the -transformation of be given by

We define two classes of sequences (see [2]).

Sequence of nonnegative numbers tending to zero is called the Rest Bounded Variation Sequence, or briefly , if it has the property

for all positive integer , where is a constant depending only on .

Sequence of nonnegative numbers will be called the Head Bounded Variation Sequence, or briefly , if it has the property

for all positive integer , or only for all if the sequence has only finite nonzero terms and the last nonzero term is .

Now, we define the other classes of sequences.

Following Leindler [3], sequence of nonnegative numbers tending to zero is called the Mean Rest Bounded Variation Sequence, or briefly , if it has the property

for all positive integer .

Analogously as in [4], sequence of nonnegative numbers will be called the Mean Head Bounded Variation Sequence, or briefly , if it has the property

for all positive integers , where the sequence has only finite nonzero terms and the last nonzero term is . It is clear that (see [5])

Consequently, we assume that the sequence is bounded, that is, that there exists a constant such that

holds for all , where denote the constants for the sequences , appearing in the inequalities (11) and (12) as .

Now we can give the conditions to be used later on. We assume that for all and

hold if belongs to and , for , respectively.

We also define two hump matrices in the following way: a lower triangular matrix   is called a maximal hump matrix if, for each , there exists integer , such that   is nondecreasing for and   is nonincreasing for , but otherwise we will have a minimal hump matrix. The hump matrices were defined and considered in [6, 7].

As a measure of approximation of by , we use the pointwise modulus of continuity of in the space defined by the formula

and the classical one

The deviation , with and otherwise, was estimated at the point as well as in the norm of by Qureshi [8] and Lal and Nigam [9]. These results were generalized by Qureshi [10]. The next generalization was obtained by Lal [11]. In the case

the deviation was estimated by Sonker and Singh [12] as follows.

Theorem 1. Let be a -periodic, Lebesgue integrable function which belongs to the -class with and . Then the degree of approximation of , the conjugate of by means of series (2), is given by

provided

where is an arbitrary positive number with and , .

In this paper we will consider the deviations and in general form. In the theorems we formulate the general conditions for the functions and the modulus of continuity obtaining the same degrees of approximation as above and sometimes essentially better one. Finally, we also give some results on norm approximation with essentially better degrees of approximation. The obtained results generalize the results from [4, 9].

We will write if there exists positive constant , sometimes depending on some parameters, such that .

2. Statement of the Results

Let

where is a positive, with , and nondecreasing continuous function.

We can now formulate our main results. At the beginning, we formulate the results on the degrees of pointwise summability of conjugate series.

Theorem 2. Let . If matrix is a maximal or minimal hump matrix with such that and

for , then

for almost all considered , where .

Theorem 3. Let with , and let satisfy

with some . If the entries of matrix satisfy condition (22) for and if matrix is a maximal or minimal hump matrix with such that , then

for almost all considered such that exists.

Next, we formulate the results on estimates of norm of the deviation considered above. In case of the deviation , let

where is positive, with , and almost nondecreasing continuous function.

Theorem 4. Let   . If the entries of matrix satisfy condition (22) for and if matrix is a maximal or minimal hump matrix with such that , then

where .

Theorem 5. Let with , where instead of satisfies (24) with some . If the entries of matrix satisfy condition (22) for and if matrix is a maximal or minimal hump matrix with such that , then

Finally, we give corollary and remarks as an application of our results.

Taking when and when with , and when and when with , Theorem 2 (Theorem 3 analogously) implies the following.

Corollary 6. If , then

for almost all considered , where , , , and .

Remark 7. In special case, if , then

and if , then

for almost all considered , where and .

Remark 8. Taking , we have, by Theorem 3 with , for and , the estimate like in [12] with the better order of approximation without any additional assumptions.

Remark 9. Analyzing the proofs of Theorems 2–5, we can deduce that, taking the assumption or instead of or , respectively, we obtain the results like that from [13].

3. Auxiliary Results

We begin this section by some notations following A. Zygmund [1, Section 5 of Chapter II].

It is clear that

where

Hence

where

Now, we formulate some estimates for the conjugate Dirichlet kernels.

Lemma 10 (see [1]). If , then

and for any real one has

Lemma 11. Let be such that (22) for holds. If , then

and if , then

with and , for .

Proof. Let

The relation implies

whence

and thus, by our assumption,

we obtain

The relation implies

whence

and thus, by our assumption,

we get

Now, our proof is complete.

Lemma 12. If matrix is a maximal or minimal hump matrix with , then

or

for , respectively.

Proof. Since the above formulas are similar, we prove the first one only. If , then

whence nondecreases with respect to and therefore

But if , then

and our proof is complete.

Lemma 13. If and , then

holds for every natural and all real .

Proof. The proof follows by the easy account

Now, our proof is complete.

4. Proofs of the Results

4.1. Proof of Theorem 2

We start with the obvious relations

By Lemmas 10 and 13, we have

for .

Using Lemmas 11 and 12 we obtain

or

whence

Collecting these estimates, we obtain the desired result.

4.2. Proof of Theorem 3

We start with the obvious relations

By the Hölder inequality , Lemma 10, (25), and (24),

We can estimate the term by the same way like in the proof of Theorem 2:

Collecting these estimates, we obtain the desired result.

4.3. Proofs of Theorems 4-5

The proofs are similar to these above and follow from the evident inequality

and additionally in case of Theorem 5 from the estimate

for .

4.4. Proof of Corollary 6

First of all we note that our matrix is the hump matrix with .

Next, we have to verify the assumptions of Lemma 11. For the first one, we note that any nondecreasing sequence belongs to the class and any nonincreasing sequence belongs to the class . The second one follows from the following calculations:

Thus proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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