ISRN Applied MathematicsVolume 2011 (2011), Article ID 725638, 20 pagesdoi:10.5402/2011/725638

Research Article

On the Favard-Type Theorem and the Jackson-Type Theorem (II)

Hee Sun Jung,¹ Ryozi Sakai,² and Noriaki Suzuki²

¹Department of Mathematics Education, Sungkyunkwan University, Seoul 110-745, Republic of Korea²Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

Received 30 September 2011; Accepted 20 October 2011

Academic Editors: C. Lu and E. Yee

Copyright © 2011 Hee Sun Jung et al. 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.

Let , and let be an even function. We consider the exponential weights , . In this paper we investigate the relations between the Favard-type inequality and the Jackson-type inequality. We also discuss the equivalence of two K-functionals and the modulus of smoothness.

1. Introduction and Preliminaries

Let , and let be an even function. We consider the weights satisfying for all , First we need the following definition from [1]. We say that is quasi-increasing if there exists such that , .

Definition 1.1. One defines, as follows: Let be continuous and an even function, and satisfy the following properties: is continuous in , with ; exists and is positive in ;the function is quasi-increasing in with , ;there exists such that Moreover, there also exists a compact subinterval of , and such that

Example 1.2. Let . If is bounded, then we call the weight the Freud-type weight. The following example is the Freud-type weight: If is unbounded, then we call the weight the Erdös-type weight. The following examples give the Erdős-type weight . ([1, Example  1.2], [2, Theorem  3.1]). For , ,where . More generally, we define for , , , and , where if , otherwise . For , .

We need the Mhaskar-Rakhmanov-Saff numbers defined by If is measurable, we define where if , then we suppose that is continuous on and The class of all functions for which will be denoted by , with the usual understanding that two functions are identified if they are equal almost everywhere. For , the degree of weighted polynomial approximation is defined by where denotes the class of all polynomials of degree at most . There are various estimates for this degree. Among them, in our previous article [3], we discuss the relation between the Favard-type inequality, the Jackson-type inequality, and the estimates by two -functionals and . We recall and summarize them in Section 2. In Section 3, we will discuss the equivalence of and modulus of smoothness . As the result, we see for a weight in a subclass of . For any nonzero real valued functions and , we write if there exists a constant independent of such that for all Similarly, for any positive numbers and we define . Throughout this paper denote positive constants independent of , or . The same symbols do not necessarily denote the same constants occurrences.

2. Known Results and Summarization

Let be an integer (in this paper, we suppose that (or ) is an integer). The th order -functional of a function is defined by the formula for , where the infimum is taken over all times continuously differentiable such that is absolutely continuous, and (this means ). Using this th order -functional, we can estimate the order of .

Theorem 2.1 (see [3]). Let . Let , be integers, and let . Let be times continuously differentiable, be absolutely continuous on each compact interval, and (when , these assumptions state merely that ). Then, for every integer ,

Theorem 2.1 was shown by using the following Favard-type inequalities (see [3]).

Theorem 2.2 ([4, Corollary 8]). Let . Let be times continuously differentiable, and let for some integer be absolutely continuous on each compact interval. Let and . Then one has equivalently,

Remark 2.3. ([4, Remark 11]) Let and let . Then there exists a constant such that for every absolutely continuous function with and , and for every , we have ([3, Section 4]) As a by-product of the method of the proof for Theorem 2.2, we can obtain the degree of functions which satisfy the Hölder-Lipschitz condition. Let and . Let be absolutely continuous with (and for , we require to be continuous, and to vanish at ). Let and set Then we define

Now, we have the following theorem:

Theorem see ([3, Theorem 4.2]). Let , , and let . Then one has

We define the following class of weights from [5, Definition  1.1].

Definition 2.5. Let , where is even, continuous, and is positive in . Then one writes , if the following are satisfied: is strictly increasing in with ;the function is quasi-increasing in for some and ;assume for some positive constants , and .

Remark 2.6. Let and is unbounded, then we see . In fact, we see this as follows.From Definition 1.1(e) and (d), we have for , Therefore we obtain (c) in Definition 2.5 for .

If , and , then we define the differences of inductively by the formula We set By [6], when is the Erdős-type weight we define for , If is the Freud-type weights, then we define

The following Jackson-type inequality is known.

Theorem 2.7 (see [5, Theorem  1.2], [6, Corollary  1.4]). Let . Let . Then for for which (and for , we require to be continuous, and to vanish at ), one has for , where , , do not depend on and .

We also consider the following class of weights which are called the Freud weights.

Definition 2.8 ([7, Definition 3.3]). Let , where is even, and exists and is positive on . Moreover, assume that is strictly increasing, with right limit 0 at 0, and for some , , Then we call Freud weight, and write .

Remark 2.9. Let . Then we see the following. If is bounded, then we say that is the Freud-type weight, and we write . Then we see . In fact, when , and large enough, by Definition 1.1(e), and then since is bounded, there exists such that Similarly, Therefore, if we take large enough, then we have (2.17).Conversely, to show we suppose that there exists such that is unbounded. Then since is quasi-increasing in , we see that as . So, for any there exists (large enough) such that for . Therefore, we have for and any , where is a constant, that is, (1) does not hold. Hence we have . Consequently, we have .If is bounded, then for , there exists such that

Theorem 2.10  2.10 ([7, Theorem 3.5]). Let . Let . Then for for which (and for , one requires to be continuous, and to vanish at ), one has for , where , , do not depend on and .

Damelin [6] introduces the following -functional: let , and be an integer, then we define where are chosen in advance and

Then Damelin gives the following.

Theorem 2.11 ([6, Theorem 1.3 (b)]). Let , , , and let (and for , one requires to be continuous, and to vanish at ). Then one has

For the Freud-type weights we have also the followings.

Theorem 2.12 ([7, Theorem 3.9, 3.10]). Let , , , and let (and for , we require to be continuous, and to vanish at ). Then we have

For , we see easily that

So, from Theorem 2.1 we obtain the following corollary.

Corollary 2.13. Let . Let be integers, and let . Let be times continuously differentiable, be absolutely continuous, and (when , these assumptions state merely that . Then, for every integer ,

The main theme in [3] is to summarize the above theorems. Let be an integer, and let . We have the following succession of the theorems. We use the constant which do not depend on and .(a)[Theorem 2.7 with (the Erdős-type case)], [Theorem 2.10 with (the Freud case)]:let and if , assume that . If , assume in addition that is continuous and that has limit 0 at . Then we have (b) [Theorem 2.2]: let be times continuously differentiable, and let for some integer , be absolutely continuous on each compact interval. Let and . Then we have equivalently, (c) [Theorem 2.1]:let be an integer, and let be times continuously differentiable, be absolutely continuous, and (when , these assumptions state merely that ). Then, for every integer and , (d) [Theorem 2.7, 2.11 (the Erdős-type case)], [Theorem 2.10, 2.12 (the Freud case)]:for with (and for we require to be continuous, and to vanish at ), for every large enough and for every integer we have

Consequently we find an interesting fact as follows:

(b) is shown by the use of (a) (see [4]). Using (b), we proved (c). If we use [6, Theorem 1.3 (b)] and [7, Theorems 3.9 and 3.10], we see easily that (c) means (d) with Theorem 2.11. Trivially, we have (a) from (d).

3. Equivalence of -Functional and Modulus of Smoothness

In this section we consider a subclass of .

Assumption 3.1. Let be an integer. Let . Then we suppose that , exists in , furthermore, the following inequalities hold where for we suppose that (3.1) hold almost everywhere on . Then we write .

Remark 3.2. Example 1.2, (i), (ii) satisfy Assumption 3.1, moreover , : a positive integer, satisfies Assumption 3.1.

In Section 2, we know The equivalences mentioned in the last of Section 2 give a certain suggestion, that is, In fact, this is true.

Theorem 3.3. Let be a positive integer, and let . Let . Then one has (3.3). Similarly, when and , one has (3.3).

To prove Theorem 3.3, we need some lemmas.

Lemma 3.4. Let be an integer. If is absolutely continuous on , then for , one has the following representation:

Proof. Therefore, for we have (3.4). For some we suppose (3.4). Then we have where Hence we have (3.4) for .

Lemma 3.5 (see [1, Lemma  3.4 ]). Uniformly for , one has

Lemma 3.6. Let and be an integer. Then for any integer , , there exist and such that for , Also, there exist and such that for , Furthermore, for , (3.9) and (3.10) hold almost everywhere on and , respectively. When and , one also has (3.9) and (3.10).

Proof. First, we will see that for the following equations hold: For we take as follows; and for all , satisfies (3.1) for all . Then we obtain that exchanging with , (3.11) also holds.Let , and let . and we continue this manner, so where are coefficients. Here, from (3.1), for large enough and , Hence, from (3.14) and (3.15) we have where is a positive constant. Therefore, we have (3.11) with . Similarly, for , Therefore, we have (3.11) for , and hence we also have (3.9) and (3.10) for large enough. If in (3.11), we replace with , then repeating the above proof we also obtain (3.11), so for we conclude (3.9) and (3.10) with large enough, that is, For , the lemma is trivial.

Lemma 3.7. Let be an integer. There exists such that for every integer , and for ,

Proof. From Definition 1.1(e) we see that there exist and such that so we have Hence, we have (3.19). From (3.9) with , we see that for and constants , Especially, if we use (3.24) with , then we have Then, for there exists a constant such that In fact, for we have that is, where . Hence we have (3.26).Now, by (3.26), if , then we have When , we see easily that Hence, with (3.29) we have for , Consequently, from (3.24) and (3.31) we have (3.20). We need to show (3.21). For large enough, we see (see Definition 1.1(e)) so we can select large enough such that We show (3.21) for , . For , we have by (3.33), Hence, we have for a certain constant , For we have (3.21) by the use of (3.19). In fact, there exists such that So, with (3.35) we conclude (3.21).

To prove Theorem 3.3 we further need the following lemma.

Lemma 3.8. For , one lets , and for integer one lets . Let , and be an integer. If is absolutely continuous, , and , then

Proof. We will prove (3.37) for and , and then we use the Riesz-Thorin interpolation theorem. Let Then for almost all , Denoting we get by changing of the integral order. Hence, from (3.20) we have By the definition of (3.38), we have (3.37) for .Next, we show (3.37) for . From (3.39) we see that by (3.21). So we have (3.37) for . Let , , then we set So we have Since is dense in , , using the Riesz-Thorin interpolation theorem for the linear operator, we have the result.

Corollary 3.9. For , one lets , and for integer one lets . Let , and be an integer. If is absolutely continuous on , , , and , then

Lemma 3.10 ([4, Lemma  7]). Let . For a certain constant , let satisfy Then there exists a constant such that for every and which satisfy and , one has

Proof of Theorem 3.3. For a given , we can select , where is absolutely continuous on , and such that Let , where . Then we have Let . From Lemma 3.4, 3.10 and the Hőlder-Minkowski inequality, we have where . We estimate Then we may suppose Using Lemma 3.5, we see (see [7, page 12]). By Corollary 3.9 with and (3.55) we have Hence, from (3.51), (3.52), and (3.56) we have Consequently, we have Therefore, from (3.2) we have (3.3).

Corollary 3.11. For , one lets , and for integer one lets . Let . Then one has

Acknowledgments

The authors thank the referees for many kind suggestions and comments. H. S. Jung was supported by SEOK CHUN Research Fund, Sungkyunkwan University, 2010.

References

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