Journal of Function Spaces and ApplicationsVolume 2012 (2012), Article ID 202987, 9 pagesdoi:10.1155/2012/202987

Research Article

On Functions of Bounded -Variation

N. Merentes,¹ S. Rivas,² and J. L. Sanchez¹

¹Universidad Central de Venezuela, Caracas, Venezuela²Universidad Nacional Abierta, Caracas, Venezuela

Received 7 April 2010; Accepted 24 November 2010

Academic Editor: Jürgen Appell

Copyright © 2012 N. Merentes 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.

We introduce and study the concept of -variation (, of a real function on a compact interval. In particular, we prove that a function has bounded -variation if and only if is absolutely continuous on and belongs to . Moreover, an explicit connection between the -variation of and the -norm of is given which is parallel to the classical Riesz formula characterizing functions in the spaces and . This may also be considered as an alternative characterization of the one variable Sobolev space .

1. Introduction

About 120 years ago, Jordan [1] introduced the notion of a function of bounded variation and the corresponding function space . He also proved the important result that if and only if with both and being monotonically increasing. Later the concept of bounded variation was generalized in various directions. In 1908, De la Vallée Poussin [2] introduced the space of functions with bounded second variation. It is known that if and only if can be represented in the form , where both and are convex.

This was further generalized by Popoviciu [3] who introduced, for any , the notion of th variation and defined the corresponding space of functions of bounded th variation on . It is known that, for any , the derivative belongs to ; therefore, there exist the right- and left-hand derivatives and on . Moreover, the set of points where does not exist is at most countable, the derivative is continuous on , and the unilateral right derivative and the unilateral left derivative are right continuous and left continuous, respectively.

In [4], the first author defined and studied the notion of the so-called bounded –variation and proved a generalization of the well known Riesz lemma. More precisely, a function has bounded -variation if and only if , where denotes the space of all absolutely continuous functions on , and . Moreover, the -variation of on is then given by the formula

In this paper we will prove the following parallel result: given and , a function has bounded -variation on if and only if and . Moreover, the -variation of on is then given by

This characterization can be considered as a natural generalization of the classical Riesz lemma [5] for the class of all functions such that and . We point out that the Riesz lemma provides a criterion for functions to belong to the Sobolev space more than 20 years before this space has been introduced by Sobolev [6] in 1934.

2. Preliminaries

In this section, we recall some definitions and known results concerning the Riesz -variation, the De la Vallée Poussin second variation, and the Popoviciu th variation.

Given a function and a partition of , consider the expression

The number , where the supremum is taken over all partitions of , is called the Riesz -variation of on . In case , the function is said to have bounded -variation. In what follows, by we will denote the class of all functions of bounded -variation on (in the Riesz sense).

The class is a Banach space with respect to the norm

The following characterization of functions is known in the literature as the Riesz lemma [5].

Proposition 2.1. A function belongs to if and only if , that is, and . Moreover, the equality holds in this case.

In 1908, De la Vallée Poussin [2] introduced the class of functions of bounded second variation as follows. Given a function and a partition of of the form consider the expression and let where the supremum is taken over all partitions of the form (2.4). The number is called the De La Vallée Poussin second variation of on . In case , the function is said to have bounded second variation. In what follows, by we will denote the class of all functions of bounded second variation on .

The following result has been proved in [7], see also [8].

Proposition 2.2. Every function is Lipschitz continuous on and can be expressed as a difference of two convex functions.

If , then from standard properties of convex functions it follows that the unilateral right derivative and the unilateral left derivative exist on . Moreover, the set of points where fails to exist is countable, is continuous on , the right unilateral derivative is right continuous, and the left unilateral derivative is left continuous on .

Finally, let us recall Popoviciu’s more general definition of bounded th variation [3]. To this end, we need the concept of the th divided difference of a function with respect to distinct points (not necessarily in increasing order) defined by

By definition, the th divided difference (2.7) is independent of the order in which the points appear. The following two results are well known (see, e.g., [3, 9]).

Proposition 2.3. Suppose that belongs to for some . Then, for some , where denotes the smallest interval containing .

Proposition 2.4. Suppose that is such that is Riemann integrable on for some . Then,

If the function has a Riemann integrable derivative of order , then the last result allows us to generalize the concept of the th divided difference for points which are not necessarily distinct. Moreover, if the unilateral derivatives and are right and left continuous on , respectively, then the function of variables is continuous in each variable separately.

As a consequence of the source Propositions 2.3 and 2.4 we have that, if for some , then

However, as a corollary to Proposition 2.4 we can obtain a weaker version of the preceding results.

Proposition 2.5. Suppose that is such that is Riemann integrable on for some . Assume, in addition, that the unilateral right derivative is right continuous and the unilateral left derivative is left continuous on . Then,

Let be a function and a partition of the form Moreover, let where the supremum is taken over all partitions of the form (2.12). The number is called the Popoviciu th variation of on . In case , the function is said to have bounded th variation on and the set of such functions is denoted by .

The following result on the regularity of higher derivatives is also well known [7].

Proposition 2.6. Suppose that belongs to for some . Then, the derivative belongs to ; therefore, the unilateral right derivative and the unilateral left derivative exist on . Moreover, the set of points where the derivative fails to exist is countable, is continuous on , the unilateral right derivative is right continuous, and the unilateral left derivative is left continuous on .

3. Main Result

In what follows, for and we denote by the class of all functions such that and . Thus, .

In this section we introduce the notion of -variation, where always denotes a real number in and denotes a natural number. For we will prove that a function has bounded -variation (for the definition see below) if and only if . Moreover, we will show that equality (1.2) holds, as one should expect.

So given , consider a partition of the form

Moreover, define where the supremum is taken over all partitions of the form (3.1). We call the number the -variation of on . In case , we say that the function has bounded -variation, and we denote the space of such functions by .

Proposition 3.1. For and , the inclusion and the inequality hold true, where is given by (2.14).

Proof. Let be a partition of of the form (3.1). By Hölder’s inequality, we obtain Passing on both sides of to the supremum over all partitions of the form (3.1), we conclude that (3.4) is true, and so also (3.3), by definition of .

From Proposition 2.6 and Proposition 3.1 we immediately deduce the following.

Corollary 3.2. Let , , and . Then, the derivative belongs to ; therefore, the unilateral right and the unilateral left derivatives and exist on and are right and left continuous, respectively.

Now we prove a statement on the existence of the derivative for all in case .

Proposition 3.3. Let , , and . Then, the derivative exists for all .

Proof. Suppose that there exists such that . Let be a partition of of the form where satisfies By definition of the -variation of , we have then From Corollary 3.2 we know that the unilateral right derivative and the unilateral left derivative are right and left continuous, respectively, on . Now, by Corollary 3.2, we further have contradicting our assumption . Consequently, the function has a derivative everywhere on .

We are now in a position to formulate and prove our main result.

Theorem 3.4. Let and . Then, if and only if . Moreover, in this case the -variation of on is given by (1.2).

Proof. Suppose first that belongs to . By Proposition 3.3, we know then that the derivative exists on .Let be a partition of . For any subinterval , , , we define a partition in the form where Using the shortcut by Corollary 3.2, we have then Consequently, Hence, by the Riesz lemma, we see that and , that is, . Moreover, Conversely, suppose now that belongs to , and let be a partition of of the form (3.10). Since and , by Proposition 2.4, we have where and . By Hölder’s inequality, we get then the estimates This implies that and Combining estimates (3.15) and (3.18), we conclude that (1.2) is true, and so the proof is complete.

The characterization of functions in included in Theorem 3.4 can be considered as a natural generalization of that given by Riesz [5] for the class , and by Merentes [4] for the class .

Acknowledgments

This paper has been partly supported by the Central Bank of Venezuela which is gratefully acknowledged. the authors also express their gratitude to the staff of the math library for compiling the references.

References

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