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Journal of Function Spaces
Volume 2014, Article ID 201801, 13 pages
http://dx.doi.org/10.1155/2014/201801
Research Article

Estimates of Modulus of Continuity of Generalized Bounded Variation Classes

1School of Mathematical Sciences, BCMIIS, Capital Normal University, Beijing 100048, China
2NO. 2 Middle School of Changping, Beijing 102208, China

Received 28 February 2014; Accepted 11 April 2014; Published 4 May 2014

Academic Editor: Józef Banaś

Copyright © 2014 Heping Wang and Zhaoyang Wu. 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

Some sharp estimates of the modulus of continuity of classes of -bounded variation are obtained. As direct applications, we obtain estimates of order of Fourier coefficients of functions of -bounded variation, and we also characterize some sufficient and necessary conditions for the embedding relations . Our results include the corresponding known results of the class as a special case.

1. Introduction and Main Results

To generalize the notion of functions of bounded variation, Wiener [1] introduced the class of functions of -bounded variation. Young [2] introduced the notion of functions of -bounded variation, and Waterman [3] studied a class of -bounded variations. Combining the notion of -bounded variation with that of -bounded variation, Leindler [4] introduced the class of functions of -bounded variation, and both classes of -bounded variation and -bounded variation are its special cases. Actually the class first appeared in Schramm and Waterman’s paper [5], and some restrictions are imposed on in their definition. Here we adopt Leindler’s definition.

Definition 1. Let be a nondecreasing function with , and let be a nondecreasing sequence of positive numbers such that . Let be the set of all sequences of nonoverlapping subintervals in . If for any , a real valued function satisfies the condition then is said to be of -bounded variation, and this fact is denoted by . And the quantity is said to be -total variation of .
In the special case when , is said to be of -bounded variation, and we write and , and if , is said to be of -bounded variation, and we denote and .
In the case , we get the class of -bounded variation, and is said to be of -bounded variation, and we denote . More specifically, when , we say that is of -bounded variation, and we denote and . The class is also called the Wiener class and is the well- known class of bounded variation .
It is easily seen from the definition that functions are bounded; that is, , and the discontinuities of a function are simple and therefore at most denumerable, where denotes the class of bounded real valued functions on .
Let be a nondecreasing sequence of positive numbers such that . If a continuous and nondecreasing function on such that , and , , then we say that generates . By the nondecreasing property of , it is easily verified that if generates , then
Let be a nonnegative real-valued function on . If there exists such that , for some positive constant , then we say that satisfies the condition . If generates and satisfies the condition , we say that satisfies the condition . Obviously the condition here is a very weak restriction on and .
Let be a modulus of continuity, that is, a continuous and nondecreasing function on satisfying and for nonnegative and . As usual, for , denote by the class of functions for which , where is the modulus of continuity of . We write instead of and instead of , the Lipschitz class, for brevity.
Functions of classes , , , and are considered in trigonometric Fourier series and some of them share good approximative properties (see [13, 611], etc.). What we mention here is the following theorem proved by Shiba [12], Schramm and Waterman [5], and Wang [13]:

Theorem A. (a) If , , , and where , then the Fourier series of converges absolutely.
(b) If is , , , , and where , then the Fourier series of converges absolutely.

Embedding relations between various generalized bounded variation classes and the class (or the Lipschitz class ) are also investigated in recent years. It is well known that . For , the estimates of modulus of continuity of had been given in [13] for and in [9, 14] for . Furthermore, Goginava in [15] and Li and Wang in [9] proved that, for , For more detailed results on this topic, we refer readers to [4, 9, 10, 1425].

In this paper, we obtain some sharp estimates of modulus of continuity of the classes in the case of that is convex. More specifically, our results include estimates of modulus of continuity of the classes , estimates of modulus of continuity of the classes , and specially estimates of modulus of the classes . Our results extend and include the corresponding known results of the class as a special case and are also sharp in most cases. As direct applications, we obtain estimates of order of Fourier coefficients of functions of -bounded variation, and we also characterize some sufficient and necessary conditions for the embedding relations and .

Now we state our main results, and in what follows, without loss of generality, we always assume that , and functions in various generalized bounded variation classes are periodic.

Theorem 2. Let be the class of functions of -bounded variation, and let generate , where is convex and is the inverse function of . Then, for , and this estimate is sharp in the sense of order provided that satisfies the condition .

Corollary 3 is a direct result of Theorem 2 by choosing and , respectively.

Corollary 3. (a) Let be the class of functions of -bounded variation, and let generate . Then, for , and this estimate is sharp in the sense of order provided that satisfies the condition .
(b) Let be the class of functions of -bounded variation, where is convex. Then, for , and this estimate is sharp in the sense of order.

The first part of Corollary 3(a) is due to Wang [13].

For special ’s, one has

Corollary 4. Let and be convex. Then(a)for , (b)for , (c)for , (d)for , (e)for , (f)for ,
The above estimates are sharp in the sense of order.

Let be an integrable function on and let its Fourier coefficients be defined as follows: We note that Theorem 2 implies the following estimates of Fourier coefficients.

Corollary 5. (a) Let be the class of functions of -bounded variation, and let generate , where is convex and is the inverse function of . Then, for ,
(b) Let be the class of functions of -bounded variation, where is convex and is the inverse function of . Then for

Corollary 5(a) includes Wang’s result [13] for the class as a special case of .

Theorem 6. Let be class of functions of -bounded variation, and let generate . Set and assume .
(a) For , and this estimate is sharp in the sense of order provided that satisfies the condition .
(b) For ,
(c) If, for some , is bounded on , then

Unfortunately we cannot assert the sharpness of our estimates in (b) and (c) of Theorem 6 for the case . Our next theorems concern some important special case of the class and elaborate on the estimates in Theorem 6. Among the classes considered here are and .

Theorem 7. (a) Let be the class of functions of -bounded variation. Then, for and , one has (i), ;(ii), .
(b) Let be the class of -bounded variation, that is, the Wiener class. Then, for and , and both estimates in (a) and (b) are sharp in the sense of order.

Li and Wang’s results in [9] are extended in the Theorems 6 and 7, which can be treated as the case of our theorems.

Theorem 8. Let . The following assertions are true.(a)For , (b)For , (c)For , (d)For , (e)For
And all above estimates are sharp in the sense of order.

Remark 9. In (e), for , we only have We do not know whether this estimate is sharp in the sense of order. However there exists such that This exception indicates that the estimates of the modulus of continuity of classes of -bounded variation are complicated, and Theorem 6 cannot cover all cases of the class .

As direct applications of the above theorems, we characterize some sufficient and necessary conditions for the embedding relations between the generalized bounded variation classes and the class (or ).

Corollary 10. On the embedding relations between the generalized bounded variation classes and the class or , the following assertions are true.(a)Let be the class of functions of -bounded variation and let be convex; let generate and satisfy the condition . Then (b)Let be the class of functions of -bounded variation and let be convex. Then (c)Let be the class of functions of -bounded variation and let generate .If, for , satisfies the condition , then If, for , is bounded on and , then .(d)Let be the class of functions of -bounded variation. Then, for , if and only if . if and only if .(e)Let be the class of functions of -bounded variation, that is, Wiener class; then, for , if and only if .

This paper is organized as follows. In Section 2, we first state three lemmas, and then by them we prove Theorem 2. Lemma 12 provides our proofs crucial upper estimates and Lemma 13 will be used repeatedly in proving the sharpness of our estimates. In Section 3, we prove Theorem 6, and the same estimate technique in [9] is partly used in our proof. Theorems 7 and 8 are proved in Section 4. For the case , the difficulty in our proofs of Theorems 7 and 8 is to prove the sharpness of our estimates and the key is to construct extreme functions by Lemma 13.

2. Proof of Theorem 2

Before we start our proof of Theorem 2, we prove three lemmas. Lemmas 11 and 12 will also be used in the proof of Theorem 6. Lemma 13 will be employed repeatedly in the proof of the sharpness of our estimates. Lemmas 12 and 13 are of independent interest for functions of -bounded variation.

Lemma 11. Let . Then is continuous on .

Proof. Using triangle inequality of norm, for any , , we have The right continuity of at implies the continuity of .

Lemma 12. Let be an arbitrary sequence of nonoverlapping intervals in , , convex, and the function of . Then, for , Specifically, if , then .

Proof. By the definition of -total variation and letting and in summation transform we have
Note that if is an increasing convex function on , then is increasing and concave on . The concavity of implies that . Therefore, by Jensen’s inequality and the above inequality, we finally get This completes the proof of Lemma 12.

Lemma 13. Let be a given set of nonnegative and nonincreasing numbers. Define a function on as and extend it to with period . Then(a) and ;(b), ;(c).

Proof. By the definition of -total variation and the nonnegative and nonincreasing properties of , direct computation proves (a). Since is nondecreasing, it is obvious that (a) implies (c). For (b), computation shows that

Now we prove Theorem 2.

Proof of Theorem 2. We write for simplicity. By Lemma 11, there exists an such that If we set and consider the periodicity of , then where From Jensen’s inequality and Lemma 12, for all , the concavity of implies Substituting (46) into (44), we get Since the right of (47) is decreasing with respect to and , the estimate in Theorem 2 is obtained from (47) directly.
Now we show that our estimate is sharp in the sense of order under the assumption that satisfies the condition .
We choose and consider function defined in Lemma 13.
We have
If satisfies the condition , that is, there exists such that , , then and this yields
From (49) and (52), we have
On the other hand, (50), the concavity and the monotonicity of imply that Obviously, (53) and (54) mean the sharpness of our estimate in Theorem 2.

Proof of Corollary 4. Obviously the ’s in Corollary 4 satisfy the condition . The proofs of (a), (b), (c), and (e) in Corollary 4 are obvious. In (d), we have and in (f), we have which complete the proofs of (d) and (f) of Corollary 4.

3. Proof of Theorem 6

In this section we prove Theorem 6. The proof of Theorem 6(a) is based on Hölder’s inequality and Lemma 12. We use techniques used by Li and Wang [9] in the proofs of (b) and (c) of Theorem 6. Lemma 12 plays a crucial role in the whole proof of Theorem 6.

Proof of Theorem 6. As in the proof of Theorem 2, it is easily seen from Lemma 11 that, for , there exists an such that where
We first prove (a). We note that for . Let satisfy . By Hölder’s inequality and Lemma 12, we have For , Lemma 12 directly yields Thus, for , we obtain Substituting (61) into (57) and noting that , we prove the desired estimate in Theorem 6(a).
If satisfies the condition , then from the proof of Theorem 2, we know that If we choose and consider the functions defined in Lemma 13, then we have Equations (64) and (65) show that the estimate in Theorem 6(a) is sharp.
Now we prove (b). Without loss of generality, we assume that and denote and . From the definition of , we first have Denote by the set of integers for which Then , and there are at most nonempty , where denotes the number of the elements in . Obviously .
Hence we have with to be determined.
From (67) and Lemma 12, we have, for , and thus Therefore, If we set , where is the inverse function of , then From the monotonicity of , it is easily verified that . And from this we also have This yields
Now we estimate and in (68).
By means of (72) and (74), we first have If we choose , then .
From (72) and the monotonicity of , we have Inserting (75) and (76) into (68) and noting that , for , we get Combining (77) and (57), we finally obtain Since and the right of (78) is decreasing with respect to , the proof of Theorem 6(b) is complete.
Finally we prove Theorem 6(c). Notice that is nondecreasing. It follows from (67) that where and . It is obvious that .
By (72) and (74), we know that Assume that is bounded on . From (72) and the fact , , we obtain Finally, (79) yields Since , we prove Theorem 6(c) from (57).

4. Proofs of Theorems 7 and 8

In this section, we prove Theorems 7 and 8.

Proof of Theorem 7. Theorem 7(a) implies Theorem 7(b) by letting . We only need to prove Theorem 7(a). Obviously satisfies the condition and we have For , it follows from Theorem 6(a) that , and the order of this estimate is sharp.
For , we have Substituting into Theorem 6(b), we get .
For , , we have By Theorem 6(c), we get the estimate
If , then satisfies the condition , and For , (ii) follows from Theorem 6(a) and the order in the estimate (ii) is also sharp.
For , we have Substituting into Theorem 6(b), we get (ii).
Now we show the sharpness of the estimates in (i) and (ii) for .
For this purpose, we consider defined in Lemma 13.
In (i), for , we choose From Lemma 13, we know that
For , , we choose where From Lemma 13, we have
In (ii), for , we choose where From Lemma 13, we know that
Obviously the above functions chosen prove the sharpness of our estimates in Theorem 7 for .

The proof of Theorem 8 is similar to that of Theorem 7, but computations are more complicated.

Proof of Theorem 8. Let generate and . It is readily seen that ’s in (a), (b), (c), (d), and (e) satisfy the condition .
In (a), we have In (b), we have In (c), we have In (d), we have In (e), we have From Theorem 6(a) we obtain sharp estimates in (a), (b), (c), (d), and (e) for .
Now we prove the estimates in Theorem 8 for . We apply (b) and (c) in Theorem 6 for upper estimates. The technique used here to show the sharpness is the same as that of the proof of Theorem 7 and the key is to choose to construct extreme functions defined in Lemma 13.
In (a), for , we have From Theorem 6(b), we obtain If we choose