#### 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 Wienerās definition of bounded -variation. There are many other generalizations of , such as bounded -variation in the sense of Young (see [1]) and Watermanā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., [1ā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 āā 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 quasinormWe 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 byFrom the inequality āā, we get that . It then follows that is a quasinorm on .

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

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

From the inequalitywe 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 āā can be expressed as the difference of two increasing functions in .

Let , , and . We setThen 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 , ,āā, and , then āā 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 āā, 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 āā as follows.

Corollary 4. Let āā. 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 1ā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 . ThenLetting , we get thatTaking the supremum over all partitions , we have for . This means that , and as . Hence, āā is complete. Theorem 1 is proved.

Proof of Theorem 2. Suppose that āā. 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 bywhere the supremum is taken over all partitions of , . It suffices to show that . For any fixed partition , we note thatwhere the supremum is taken over all partitions of . It follows thatwhich implies . This completes the proof of Theorem 2.

To prove Theorem 3, we introduce the next lemma.

Lemma 5. If āā, 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 thatTaking the supremum over all partitions of , we obtain thatOn the other hand, for any fixed , by renumbering if necessary, we may assume that . We set āā. Then is a partition of . It follows thatLetting , we getCombining (19) with (21), we get (9). Hence, āā if and only if .
(2) Let be an increasing function in āā and the set of points of discontinuity of on . We set , where is defined by (7). Similar to the proof of (21), we haveApplying the above proved result, we obtain that . We set ; then . We will show that is continuous on .
Indeed, for , we haveBy Weierstrass -test (see [10, Theorem 7.10]), we get that the series converges uniformly on . For , āā 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.