Properties of Functions in the Wiener Class for

Abstract

We will investigate properties of functions in the Wiener class with . We prove that any function in can be expressed as the difference of two increasing functions in . We also obtain the explicit form of functions in and show that their derivatives are equal to zero a.e. on .

1. Introduction

Let . We say that a real valued function on is of bounded -variation and is denoted by , if where the supremum is taken over all partitions . When , we get the well-known Jordan bounded variation ; and when , we get Wiener’s definition of bounded -variation. There are many other generalizations of , such as bounded -variation in the sense of Young (see [1]) and Waterman’s -bounded variation (see [2]). The class and generalizations of have been studied mainly because of their applicability to the theory of Fourier series and some good approximative properties (see, e.g., [1–7]).

However, it should be mentioned that results of most papers deal mostly with the case . This is because that in this case is a Banach space with the norm (see, e.g., [3]). In the case , is no longer a Banach space and has not been studied as far as we know. Nevertheless, functions in    have many interesting properties; for example, their derivatives are equal to zero a.e. on .

In this paper, we will investigate properties of functions in the class with . We will show that is a Frechet space with the quasinorm We will get the Jordan type decomposition theorem which says that any function in can be expressed as the difference of two increasing functions in . We also get the representation theorem which gives the explicit form of functions in .

2. Statement of Main Results

Clearly, for any fixed , the Wiener class is a linear space. We define the functional on by From the inequality   , we get that . It then follows that is a quasinorm on .

Our first result claims that    equipped with the quasinorm is a Frechet space.

Theorem 1. The Wiener class    equipped with the quasinorm is a Frechet space.

From the inequality we get that, for any , which means that . Specially, for , . This implies that functions are bounded, and the discontinuities of a function are simple and, therefore, at most denumerable (see [8, Theorem 13.7 and Lemma 13.2]). By the Jordan decomposition theorem, we know that every function in can be expressed as the difference of two increasing functions and defined on (see [8, Corollary 13.6]). If , we can require that the above increasing functions and are still in . This is our next theorem.

Theorem 2 (Jordan type decomposition theorem). Any function in    can be expressed as the difference of two increasing functions in .

Let , , and . We set Then is increasing on with only one discontinuity point . Also, for .

Let be an increasing function in . Denote by the set of points of discontinuity of . Then is at most countable (see [8, Theorem 2.17]). Since is increasing, we get that, for any , the right and left limits and of the function at exist, , and . For , we define

Our next theorem characterizes the form of an increasing function in . Any increasing function in must be as follows: where , , , , and .

Theorem 3. (1) If , where , ,  , and , then    if and only if . In this case, (2) Let be an increasing function in . Then , where is a constant, is the set of points of discontinuity of , and is defined by (7).

Finally, for an increasing function in   , by Theorem 3 we have , where is the set of points of discontinuity of and at most countable. Since , a.e. , by the Fubini term by term differentiation theorem (see [9, Proposition 4.6]), we get , a.e. . By Theorem 2, any function in can be expressed as the difference of two increasing functions and in . Applying Theorem 3, we get the representation theorem of functions in    as follows.

Corollary 4. Let   . Then can be expressed in the following form: where is a constant, , are increasing functions in , and are defined by (7), , and , , are the sets of points of discontinuity of , , and , respectively. Furthermore, , a.e. .

3. Proofs of Theorems 1–3

Proof of Theorem 1. It suffices to prove that is complete. Let be a Cauchy sequence in ; that is, as . For any , using the partition and the definition of , we get that is a Cauchy sequence in and converges to a number denoted by . For any , there exists an integer such that for . Let be an arbitrary partition of . Then Letting , we get that Taking the supremum over all partitions , we have for . This means that , and as . Hence,    is complete. Theorem 1 is proved.

Proof of Theorem 2. Suppose that   . Since , by the Jordan decomposition theorem (see [8, Corollary 13.6]), we have , where , are increasing functions on . Indeed, we can choose to be , the total variation function of defined by where the supremum is taken over all partitions of , . It suffices to show that . For any fixed partition , we note that where the supremum is taken over all partitions of . It follows that which implies . This completes the proof of Theorem 2.

To prove Theorem 3, we introduce the next lemma.

Lemma 5. If   , then is a constant function.

Proof. It suffices to show that, for any , . Assume that there exists such that . Without loss of generality, we assume that . Since , there exist points such that and . Hence, as , which implies that . This leads to a contradiction. Lemma 5 is proved.

Proof of Theorem 3. (1) Without loss of generality, we may assume that . Let be a partition of . For , , we note that where an empty sum denotes 0. It follows that Taking the supremum over all partitions of , we obtain that On the other hand, for any fixed , by renumbering if necessary, we may assume that . We set   . Then is a partition of . It follows that Letting , we get Combining (19) with (21), we get (9). Hence,    if and only if .
(2) Let be an increasing function in    and the set of points of discontinuity of on . We set , where is defined by (7). Similar to the proof of (21), we have Applying the above proved result, we obtain that . We set ; then . We will show that is continuous on .
Indeed, for , we have By Weierstrass -test (see [10, Theorem 7.10]), we get that the series converges uniformly on . For ,    is continuous at , so is also continuous at . It follows that is continuous at for .
For , we set . Then is continuous at and . Hence, Thus, from which we can deduce that is continuous at . Hence, .
Since , it follows from Lemma 5 that is a constant . Thus . The proof of Theorem 3 is complete.

Conflict of Interests

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

Acknowledgments

The authors were supported by the National Natural Science Foundation of China (Project no. 11271263), the Beijing Natural Science Foundation (1132001), and BCMIIS.