Journal of Mathematics

Volume 2013 (2013), Article ID 426347, 8 pages

http://dx.doi.org/10.1155/2013/426347

Research Article

Pointwise Analog of the Stečkin Approximation Theorem

Włodzimierz Łenski

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

Received 11 November 2012; Accepted 20 January 2013

Academic Editor: Roberto Renò

Copyright © 2013 Włodzimierz Łenski. 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 show the pointwise version of the Stečkin theorem on approximation by de la Vallée-Poussin means. The result on norm approximation is also derived.

1. Introduction

Let be the class of all -periodic real-valued functions integrable in the Lebesgue sense with th power (continuous) over , and let when or when .

Let us define the norms of as

where

We note additionally that

Consider the trigonometric Fourier series of

with the partial sums .

Let

As a measure of approximation by the previous quantities, we use the pointwise characteristics

compared to [1], and

and also

where

constructed on the base of definition of -points  (Lebesgue points (-points) or points of continuity (-points)). We also use the modulus of continuity of in the space defined by the formula

and its arithmetic mean

We can observe that, for and ,

whence

whence

Let us introduce one more measure of pointwise approximation analogical to the best approximation of function by trigonometric polynomials of the degree at most

Namely,

We will also use its arithmetic mean

Using these characteristics, we will show the pointwise version of the Stečkin [2] generalization of the Fejér-Lebesgue theorem. As a corollary, we will obtain the mentioned original result of Stečkin on norm approximation as well as the result of Tanović-Miller [3].

By we will designate either an absolute constant or a constant depending on some parameters, not necessarily the same of each occurrence.

2. Statement of the Results

At the beginning, we formulate the partial solution of the considered problem.

Theorem 1 (see [4]). If , then, for any positive integer and all real ,

Now, we can present the main result on pointwise approximation.

Theorem 2. If , then, for any positive integer and all real ,

Remark 3. Theorem 2, in case , immediately yields the result of Stečkin [2]. The result of Stečkin type also holds if instead of we consider the spaces with . Then, in the proof, we need the Hardy-Littlewood estimate of the maximal function.

At every point of

and thus from Theorem 1, we obtain the corollary which states the result of the Tanović-Miller type [3].

Corollary 4. If , then, for any positive integer at every -point   of ,

3. Auxiliary Results

In order to prove our theorems, we require some lemmas.

Lemma 5. If is the trigonometric polynomial of the degree at most of the best approximation of with respect to the norm , then it is also the trigonometric polynomial of the degree at most of the best approximation of with respect to the norm for any .

Proof. From the inequalities

where and are the trigonometric polynomials of the degree at most of the best approximation of with respect to the norms and , respectively, we obtain relation

whence for any by uniqueness of the trigonometric polynomial of the degree at most of the best approximation of with respect to the norm (e.g. see [5] p. 96). We can also observe that for such and any ,

Hence,

and our proof is complete.

Lemma 6. If and , then is nonincreasing function of and nondecreasing function of . These imply that for the function is nonincreasing function of and simultaneously.

Proof. The first part of our statement follows from the property of the norm and supremum. The second part is a consequence of the calculation

Lemma 7. Let such that and . If , then

Proof. It is clear that

where

Hence, by orthogonality of the trigonometric system,

with trigonometric polynomial of the degree at most of the best approximation of .

Using the notations

we get

We next evaluate the sums and using the partial integrating and Lemma 5. Thus,

which proves Lemma 7.

Before formulating the next lemmas, we define a new difference. Let and . Denote that

Lemma 8. Let such that . If , then

Proof. The proof follows by the method of Leindler [6]. Namely,

By Lemma 6, for ,

and our proof is complete.

Lemma 9. Let and . If , then

Proof. Our proof runs parallel with the proof of Theorem 1 in [2].

If , then

and if , then

by Lemmas 6 and 7.

Next, we construct the same decreasing sequence of integers that was given by Stečkin. Let

where denotes the integral part of . It is clear that there exists smallest index such that and

By the definition of the numbers , we have

whence

follow.

Under these notations, we get the following equality:

whence, by ,

follows.

It is easy to see that the terms in the sum , by Lemma 8, with and do not exceed

and by Lemma 7, we get

Thus,

whence, by the monotonicity of ,

4. Proofs of the Results

Proof of Theorem 2. The proof follows the lines of the proofs of Theorem 4 in [2] and Theorem in [6]. Therefore, let and be fixed. Let us define an increasing sequence of indices introduced by Stečkin in the following way. Set . Assuming that the numbers are already defined and , we define as follows. Let denote the smallest natural number such that

According to the magnitude of , we define

If , we continue the procedure, and if once , then we stop the construction and define .

By the previous definition of , we have the following obvious properties:

and relations

whenever .

Let us start with

Using Theorem 1 and that , we get

The estimate of the sum in the right hand side of (55) we derive from the following one

We split the proof of this inequality in two parts. If  , then by Lemma 9,

If , then, by Lemma 7,

and since , we have

Consequently,

Since for all , changing the order of summation, we get

Using the inequality

we obtain

where denotes the smallest index having the property . Hence,

and our proof follows.

References

S. Aljančić, R. Bojanić, and M. Tomić, “On the degree of convergence of Fejéer-Lebesgue sums,” L'Enseignement Mathématique. Revue Internationale. IIe Série, vol. 15, pp. 21–28, 1969.

S. B. Stečkin, “On the approximation of periodic functions by de la Vallée Poussin sums,” Analysis Mathematica, vol. 4, no. 1, pp. 61–74, 1978.

N. Tanović-Miller, “On some generalizations of the Fejér-Lebesgue theorem,” Unione Matematica Italiana. Bollettino. B. Serie 6, vol. 1, no. 3, pp. 1217–1233, 1982.

W. Łenski, “Pointwise approximation by de la Vallée-Poussin means,” East Journal on Approximations, vol. 14, no. 2, pp. 131–136, 2008.

P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Academic Press, New York, NY, USA, 1971.

L. Leindler, “Sharpening of Stečkin's theorem to strong approximation,” Analysis Mathematica, vol. 16, no. 1, pp. 27–38, 1990.