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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 421328, 10 pages
http://dx.doi.org/10.1155/2013/421328
Research Article

Positive Interpolation Operators with Exponential-Type Weights

1Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea
2Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Received 26 December 2012; Accepted 7 March 2013

Academic Editor: Roberto Barrio

Copyright © 2013 Hee Sun Jung and Ryozi Sakai. 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 consider positive operators on the real line with property of interpolation, and we show the weighted -convergence of the operators. We will construct an analogical operator of one which is studied by Knopfmacher (1986). Furthermore, we treat the Shepard-type interpolatory operator (cf. Xie et al. (1998)).

1. Introduction

In this paper, we consider two interpolatory positive operators. For and , we construct an operator The details will be stated later, and the result is written in Section 2. Knopfmacher [1] studied the positive operator and for , he obtained a certain weighted-convergence theorem on the compact interval . The operators (1) and (2) have the property of Hermite-Fejér interpolation, that is, We also treat the interpolatory positive operator of Shepard-type. Let us define for by The operator is linear and positive, furthermore it interpolates at the zeros . In fact, we see that

The related theorem is written in Section 4.

First we need the following definition from [2]. We say that is quasi-increasing (quasi-decreasing) if there exists such that (), .

Definition 1. Let be an even function and satisfying the following properties.(a) is continuous in , with . (b) exists and is positive in . (c). (d)The function
is quasi-increasing in , with (e) There exists such that Then, we write . If there also exist a compact subinterval of and such that then we write .

Example 2. There are some typical examples of satisfying . (1) If is bounded, then the weight is the so-called the Freud-type weight. Then the typical Freud-type example would be (2) If is unbounded, then the weight is called the Erdös-type weight. Erdös-type examples are as follows. (a) (see [2, Example 1.2], [3, Theorem 3.1]) For . where More precisely, we define for , , and , where if , otherwise (but, note that gives a Freud-type weight).(b) (see [3, Theorem 3.5]) For , put .

We construct the orthonormal polynomials of degree for , that is, Let . The Fourier-type series of is defined by We denote the partial sum of by If we use the Christoffel-Darboux formula, then we obtain Here, where . The polynomials of degree are denoted by . We define the Christoffel numbers by then we have We denote the zeros of the orthonormal polynomial by . Then we define the Christoffel numbers such as .

2. Preliminaries and Theorems

We need the Mhaskar-Rakhmanov-Saff number ; We define where Moreover, we define a function for and where . For the Freud-type weight we suppose to hold as . If is the Erdös-type weight, then it always holds. So for the Freud-type weight we need to limit slightly the weights.

To state our main result, we assume some conditions for as follows.(1) is even, positive, and quasi-decreasing on . (2) for and (3) is bounded on for .

Let be the zeros of the orthonormal polynomial . Then we define the operator by (1) with , , and for each we define a pointwise modulus of continuity . When is uniformly continuous on , we set

Then our first theorem is as follows.

Theorem 3. Let , and let as . Let . Then we have the following.(a) For , and for (b) Let and be an integrable function satisfying the following condition: Then one has for being uniformly continuous and bounded on where are defined in (26).

We prepare some lemmas for the proof of the theorem.

Lemma 4. Let .(1) (see [2, Lemma 3.5 (3.27)–(3.29)]) For fixed and uniformly for , Moreover, (2) (see [2, Lemma 3.4 (3.18),(3.17), Lemma 3.8 (3.42)]) and for , where is defined in Definition 1.
(3) (see [2, Lemma 3.11 (a), (b)]) Given fixed , one has uniformly for , (4) (see [2, Lemma ]) For some , and for large enough , (5) (see [2, Lemma 3.8 (a)]) For ,

Lemma 5 ([4, Theorem 2.7]). There exists such that

Lemma 6. Let . (1) Let be the zero of . Then for and , (2) For and ,

Proof. (1) This follows from [2, Corollary 13.4, Theorem 5.7 (b)].
(2) Recall the definition of in (21). We have Hence, if , then we see Here we see because of . (see Lemma 4 (3), (4)). Therefore, we have Let . Then we see Therefore we see

Lemma 7 ([2, Theorem ]). If , then

Lemma 8. Let . Then the following results hold.
(a) For ,
(b) For

Proof. From [2, Theorem 9.3], we have the following.
(1) Uniformly for and , we have
(2) Moreover, uniformly for and , Since , we have the following results.

3. Proof of Theorem 3

To estimate the difference , we split into two parts.

To prove the theorem we start the estimation of the denominator for the operator . We will need it in Step 4.

Step 1. Let Then we have the following.

Lemma 9. There exists such that uniformly, for ,

Proof. By Lemma 7, if , , , then Since , we see and this implies that Therefore, from (41) and (52) we can obtain Using the fact (see the definition of ), we have by (41) and (52) In another case, that is, when , we also have the same result.

Step 2. Let . Let be uniformly continuous and bounded on , and let . Then we have Now, let We have the following estimation.

Lemma 10. For ,

Proof. By (61), because we know from the definition of in (54) that

Step 3. Next, we estimate . Let and let . To do so, we prepare the following. By Lemma 6,
From the property of the modulus of continuity we have, for , where is defined in (25) as uniformly in as .

We have the following estimate.

Lemma 11. For any , Then

Proof. because we know from the definition of in (54) that

Step 4. Let . Using the result of Step 1, we have the following estimate.

Lemma 12. For any , one sets Then for , and for ,

Proof. First, let . Then using (52), (66), and Lemma 9, we have From the fact that is bounded (recall the definition of ), we can continue as Then by (25) and (40), Next, suppose . Then since we have from Lemma 11,

Step 5. Using (67) and Lemmas 11 and 12, we can estimate the part as follows:
Then for and for , Therefore, with Lemma 10 we have the following result.

Lemma 13. For , and for

Proof of Theorem 3. (a) follows from Lemma 13. We will show (b). Let . Then since we know that and so for all , we have

Example 14. Let and where Then the condition (30) is satisfied.

4. Shepard-Type Operator

Let us define the positive interpolatory operator (4) for and the zeros of the orthonormal polynomial .

Let

Lemma 15 ([5, Lemma 3.3]). For , one has

Assumption 1. We suppose that, for each , where is a constant depending only on .

Remark 16. Let , and let us define If , then we say that the weight is regular. The regular weights satisfy the condition (90) (see [6, Corollry 5.5]). All weights in Example 2 are regular weights.

Lemma 17 ([3, Theorem 1.6]). Let , and let be defined by (21). Then there exists such that for every where is defined in Definition 1 (d). In particular, for the weight one has . Furthermore, if is an Erdös-type, then for any , there exists such that, for every ,

For each let us set

Our second theorem is as follows.

Theorem 18. Let be uniformly continuous on and let . Assume is a nonnegative and decreasing function with . Then one has for the Erdös-type weights, where is defined in (94).

For the Freud weights we have the following. For , let us set and (note (92) and (94)).

Corollary 19. Let , where is defined in Definition 1 (d), and let . Then, for the Freud-type weights, (95) holds with . In particular, when , one can take .

Remark 20. For the Freud-type weights we see . If we assume (90), then for the Erdös-type weights, from Lemma 17 (93), we also have .

Proof of Theorem 18. Let . We see that Let or . Then, we see where is defined in (94). If , then we have Let Then we see that . Now, we will estimate . We see that Hence we have Using for we see that Therefore, we have Then, with (102) we see Hence, using in (94), we have that, for , Consequently, with we have

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