Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 202987, 9 pages
http://dx.doi.org/10.1155/2012/202987
Research Article

On Functions of Bounded (𝑝,𝑘)-Variation

1Universidad Central de Venezuela, Caracas, Venezuela
2Universidad 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 (𝑝,𝑘)-variation (1<𝑝<, 𝑘) of a real function on a compact interval. In particular, we prove that a function 𝑢[𝑎,𝑏] has bounded (𝑝,𝑘)-variation if and only if 𝑢(𝑘1) is absolutely continuous on [𝑎,𝑏] and 𝑢(𝑘) belongs to 𝐿𝑝[𝑎,𝑏]. Moreover, an explicit connection between the (𝑝,𝑘)-variation of 𝑢 and the 𝐿𝑝-norm of 𝑢(𝑘) is given which is parallel to the classical Riesz formula characterizing functions in the spaces 𝑅𝑉𝑝[𝑎,𝑏] and 𝐴𝑝[𝑎,𝑏]. This may also be considered as an alternative characterization of the one variable Sobolev space 𝑊𝑘𝑝[𝑎,𝑏].

1. Introduction

About 120 years ago, Jordan [1] introduced the notion of a function of bounded variation and the corresponding function space BV[𝑎,𝑏]. He also proved the important result that 𝑢BV[𝑎,𝑏] if and only if 𝑢=𝑓𝑔 with both 𝑓 and 𝑔 being monotonically increasing. Later the concept of bounded variation was generalized in various directions. In 1908, De la Vallée Poussin [2] introduced the space BV2[𝑎,𝑏] of functions with bounded second variation. It is known that 𝑢BV2[𝑎,𝑏] if and only if 𝑢 can be represented in the form 𝑢=𝑓𝑔, where both 𝑓 and 𝑔 are convex.

This was further generalized by Popoviciu [3] who introduced, for any 𝑘, the notion of 𝑘th variation and defined the corresponding space BV𝑘[𝑎,𝑏] of functions 𝑢[𝑎,𝑏] of bounded 𝑘th variation on [𝑎,𝑏]. It is known that, for any 𝑢BV𝑘[𝑎,𝑏], the derivative 𝑢(𝑘2) belongs to BV2[𝑎,𝑏]; therefore, there exist the right- and left-hand derivatives 𝑢+(𝑘1) and 𝑢(𝑘1) on [𝑎,𝑏]. Moreover, the set 𝐸 of points 𝑥 where 𝑢(𝑘1)(𝑥) does not exist is at most countable, the derivative 𝑢(𝑘1) is continuous on [𝑎,𝑏]𝐸, and the unilateral right derivative 𝑢+(𝑘1) and the unilateral left derivative 𝑢(𝑘1) are right continuous and left continuous, respectively.

In [4], the first author defined and studied the notion of the so-called bounded (𝑝,2)–variation (1<𝑝<) and proved a generalization of the well known Riesz lemma. More precisely, a function 𝑢[𝑎,𝑏] has bounded (𝑝,2)-variation if and only if 𝑢AC[𝑎,𝑏], where AC[𝑎,𝑏] denotes the space of all absolutely continuous functions on [𝑎,𝑏], and 𝑢𝐿𝑝[𝑎,𝑏]. Moreover, the (𝑝,2)-variation of 𝑢 on [𝑎,𝑏] is then given by the formula 𝑉(𝑝,2)[]𝑢(𝑢;𝑎,𝑏)=𝑝𝐿𝑝.(1.1)

In this paper we will prove the following parallel result: given 𝑝(1,) and 𝑘, a function 𝑢[𝑎,𝑏] has bounded (𝑝,𝑘)-variation on [𝑎,𝑏] if and only if 𝑢(𝑘1)AC[𝑎,𝑏] and 𝑢(𝑘)𝐿𝑝[𝑎,𝑏]. Moreover, the (𝑝,𝑘)-variation of 𝑢 on [𝑎,𝑏] is then given by 𝑉(p,𝑘)[]𝑢(𝑢;𝑎,𝑏)=(𝑘)𝑝𝐿𝑝((𝑘1)!)𝑝.(1.2)

This characterization can be considered as a natural generalization of the classical Riesz lemma [5] for the class 𝐴𝑝[𝑎,𝑏] of all functions 𝑢[𝑎,𝑏] such that 𝑢AC[𝑎,𝑏] and 𝑢𝐿𝑝[𝑎,𝑏]. We point out that the Riesz lemma provides a criterion for functions to belong to the Sobolev space 𝑊1𝑝[𝑎,𝑏] 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 𝑝-variation, the De la Vallée Poussin second variation, and the Popoviciu 𝑘th variation.

Given a function 𝑢[𝑎,𝑏] and a partition 𝜋={𝑎=𝑡0<𝑡1<<𝑡𝑚=𝑏} of [𝑎,𝑏], consider the expression 𝜎𝑝(𝑢;𝜋)=𝑚𝑗=1||𝑢𝑡𝑗𝑡𝑢𝑗1||𝑝||𝑡𝑗𝑡𝑗1||𝑝1.(2.1)

The number 𝑉𝑝(𝑢;[𝑎,𝑏])=sup𝜋𝜎𝑝(𝑢;𝜋), where the supremum is taken over all partitions 𝜋 of [𝑎,𝑏], is called the Riesz 𝑝-variation of 𝑢 on [𝑎,𝑏]. In case 𝑉𝑝(𝑢;[𝑎,𝑏])<, the function 𝑢 is said to have bounded 𝑝-variation. In what follows, by 𝑅𝑉𝑝[𝑎,𝑏] we will denote the class of all functions 𝑢[𝑎,𝑏] of bounded 𝑝-variation on [𝑎,𝑏] (in the Riesz sense).

The class 𝑅𝑉𝑝[𝑎,𝑏] is a Banach space with respect to the norm 𝑢𝑝||||+𝑉=𝑢(𝑎)𝑝[])(𝑢;𝑎,𝑏1/𝑝.(2.2)

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

Proposition 2.1. A function 𝑢 belongs to 𝑅𝑉𝑝[𝑎,𝑏](1<𝑝<) if and only if 𝑢𝐴𝑝[𝑎,𝑏], that is, 𝑢AC[𝑎,𝑏] and 𝑢𝐿𝑝[𝑎,𝑏]. Moreover, the equality 𝑉𝑝[](𝑢;𝑎,𝑏)=𝑢𝑝𝐿𝑝(2.3) 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 𝑢[𝑎,𝑏] and a partition 𝜋 of [𝑎,𝑏] of the form 𝑎=𝑎1<𝑐1𝑑1<𝑏1𝑎2<<𝑏𝑚1𝑎𝑚<𝑐𝑚𝑑𝑚<𝑏𝑚=𝑏,(2.4) consider the expression 𝜎2(𝑢;𝜋)=𝑚𝑗=1||||𝑢𝑏𝑗𝑑𝑢𝑗𝑏𝑗𝑑j𝑢𝑐𝑗𝑎𝑢𝑗𝑐𝑗𝑎𝑗||||(2.5) and let 𝑉2[](𝑢;𝑎,𝑏)=sup𝜋𝜎2(𝑢;𝜋),(2.6) where the supremum is taken over all partitions 𝜋 of the form (2.4). The number 𝑉2(𝑢;[𝑎,𝑏]) is called the De La Vallée Poussin second variation of 𝑢 on [𝑎,𝑏]. In case 𝑉2(𝑢;[𝑎,𝑏])<, the function 𝑢 is said to have bounded second variation. In what follows, by BV2[𝑎,𝑏] we will denote the class of all functions 𝑢[𝑎,𝑏] of bounded second variation on [𝑎,𝑏].

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

Proposition 2.2. Every function 𝑢BV2[𝑎,𝑏] is Lipschitz continuous on [𝑎,𝑏] and can be expressed as a difference of two convex functions.

If 𝑢BV2[𝑎,𝑏], then from standard properties of convex functions it follows that the unilateral right derivative 𝑢+ and the unilateral left derivative 𝑢 exist on [𝑎,𝑏]. Moreover, the set 𝐸 of points 𝑥 where 𝑢(𝑥) fails to exist is countable, 𝑢 is continuous on [𝑎,𝑏]𝐸, the right unilateral derivative 𝑢+ is right continuous, and the left unilateral derivative 𝑢 is left continuous on [𝑎,𝑏].

Finally, let us recall Popoviciu’s more general definition of bounded 𝑘th variation [3]. To this end, we need the concept of the 𝑘th divided difference of a function 𝑢[𝑎,𝑏] with respect to distinct points 𝑡0,𝑡1,,𝑡𝑘[𝑎,𝑏] (not necessarily in increasing order) defined by 𝑢𝑡0,𝑡1,,𝑡𝑘=𝑘𝑗=0𝑢𝑡𝑗𝑡𝑗𝑡0𝑡𝑗𝑡𝑗1𝑡𝑗𝑡𝑗+1𝑡𝑗𝑡𝑘.(2.7)

By definition, the 𝑘th divided difference (2.7) is independent of the order in which the points 𝑡0,𝑡1,,𝑡𝑘 appear. The following two results are well known (see, e.g., [3, 9]).

Proposition 2.3. Suppose that 𝑢[𝑎,𝑏] belongs to 𝐶𝑘[𝑎,𝑏] for some 𝑘1. Then, 𝑢𝑡0,𝑡1,,𝑡𝑘𝑢=(𝑘)(𝜉)𝑘!(2.8) for some 𝜉conv{𝑡0,𝑡1,,𝑡𝑘}, where conv𝑀 denotes the smallest interval containing 𝑀.

Proposition 2.4. Suppose that 𝑢[𝑎,𝑏] is such that 𝑢(𝑘) is Riemann integrable on [𝑎,𝑏] for some 𝑘1. Then, 𝑢𝑡0,𝑡1,,𝑡𝑘=10𝑥10𝑥𝑘10𝑢(𝑘)𝑥𝑘𝑡𝑘𝑡𝑘1++𝑥1𝑡1𝑡0+𝑡0𝑑𝑥𝑘𝑑𝑥1.(2.9)

If the function 𝑢 has a Riemann integrable derivative of order 𝑘1, then the last result allows us to generalize the concept of the 𝑘th divided difference for points 𝑡0,𝑡1,,𝑡𝑘 which are not necessarily distinct. Moreover, if the unilateral derivatives 𝑢+(𝑘1) and 𝑢(𝑘1) are right and left continuous on [𝑎,𝑏], respectively, then the function of 𝑘+1 variables (𝑡0,𝑡1,,𝑡𝑘)𝑢[𝑡0,𝑡1,,𝑡𝑘] is continuous in each variable separately.

As a consequence of the source Propositions 2.3 and 2.4 we have that, if 𝑢𝐶𝑘[𝑎,𝑏] for some 𝑘1, then lim0+𝑢𝑡0,,𝑡0,𝑡0=𝑢+(𝑘)𝑡0𝑘!=lim0+𝑢𝑡0,𝑡0,,𝑡0.(2.10)

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

Proposition 2.5. Suppose that 𝑢[𝑎,𝑏] is such that 𝑢(𝑘) is Riemann integrable on [𝑎,𝑏] for some 𝑘1. Assume, in addition, that the unilateral right derivative 𝑢+(𝑘1) is right continuous and the unilateral left derivative 𝑢(𝑘1) is left continuous on [𝑎,𝑏]. Then, lim0+𝑢𝑡0,,𝑡0,𝑡0=𝑢++(𝑘)𝑡0𝑘!𝑎<𝑡0,<𝑏lim0+𝑢𝑡0,𝑡0,,𝑡0=𝑢(𝑘)𝑡0𝑘!𝑎<𝑡0.<𝑏(2.11)

Let 𝑢[𝑎,𝑏] be a function and 𝜋 a partition of the form 𝑎=𝑡1,1<𝑡1,2<<𝑡1,𝑘𝑡1,𝑘+1<<𝑡1,2𝑘<𝑡2,𝑘+1<<𝑡2,2𝑘<𝑡3,1<<𝑡𝑗,1<<<𝑡𝑗,𝑘𝑡𝑗,𝑘+1<<𝑡𝑗,2𝑘<𝑡𝑚1,2𝑘𝑡𝑚1<<𝑡𝑚,𝑘𝑡𝑚,𝑘+1<<𝑡𝑚,2𝑘=𝑏.(2.12) Moreover, let 𝜎𝑘(𝑢;𝜋)=𝑚𝑗=1||𝑢𝑡𝑗,𝑘+1,,𝑡𝑗,2𝑘𝑡𝑢𝑗,1,,𝑡𝑗,𝑘||,(2.13)𝑉𝑘[](𝑢;𝑎,𝑏)=sup𝜋𝜎𝑘(𝑢;𝜋),(2.14) where the supremum is taken over all partitions 𝜋 of the form (2.12). The number 𝑉𝑘(𝑢;[𝑎,𝑏]) is called the Popoviciu 𝑘th variation of 𝑢 on [𝑎,𝑏]. In case 𝑉𝑘(𝑢;[𝑎,𝑏])<, the function 𝑢 is said to have bounded 𝑘th variation on [𝑎,𝑏] and the set of such functions is denoted by BV𝑘[𝑎,𝑏].

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

Proposition 2.6. Suppose that 𝑢[𝑎,𝑏] belongs to BV𝑘[𝑎,𝑏] for some 𝑘2. Then, the derivative 𝑢(𝑘2) belongs to BV2[𝑎,𝑏]; therefore, the unilateral right derivative 𝑢+(𝑘1) and the unilateral left derivative 𝑢(𝑘1) exist on [𝑎,𝑏]. Moreover, the set 𝐸 of points 𝑥 where the derivative 𝑢(𝑘1)(𝑥) fails to exist is countable, 𝑢(𝑘1) is continuous on [𝑎,𝑏]𝐸, the unilateral right derivative 𝑢+(𝑘1) is right continuous, and the unilateral left derivative 𝑢(𝑘1) is left continuous on [𝑎,𝑏].

3. Main Result

In what follows, for 1𝑝< and 𝑘 we denote by 𝐴(𝑝,𝑘)[𝑎,𝑏] the class of all functions 𝑢[𝑎,𝑏] such that 𝑢(𝑘1)AC[𝑎,𝑏] and 𝑢(𝑘)𝐿𝑝[𝑎,𝑏]. Thus, 𝐴(𝑝,1)[𝑎,𝑏]=𝐴𝑝[𝑎,𝑏].

In this section we introduce the notion of (𝑝,𝑘)-variation, where 𝑝 always denotes a real number in [1,) and 𝑘1 denotes a natural number. For 𝑝>1 we will prove that a function 𝑢[𝑎,𝑏] has bounded (𝑝,𝑘)-variation (for the definition see below) if and only if 𝑢𝐴(𝑝,𝑘)[𝑎,𝑏]. Moreover, we will show that equality (1.2) holds, as one should expect.

So given 𝑢[𝑎,𝑏], consider a partition 𝜋 of the form 𝑎=𝑡1,1<𝑡1,2<<𝑡1,𝑘𝑡1,𝑘+1<<𝑡1,2𝑘<𝑡2,𝑘+1<<𝑡2,2𝑘<𝑡3,1<<𝑡𝑗,1<<𝑡𝑗,𝑘𝑡𝑗,𝑘+1<<𝑡𝑗,2𝑘<𝑡𝑚1,2𝑘𝑡𝑚1<<𝑡𝑚,𝑘𝑡𝑚,𝑘+1<<𝑡𝑚,2𝑘=𝑏.(3.1)

Moreover, define 𝜎(𝑝,𝑘)(𝑢;𝜋)=𝑚𝑗=1||𝑢𝑡𝑗,𝑘+1,,𝑡𝑗,2𝑘𝑡𝑢𝑗,1,,𝑡𝑗,𝑘||𝑝||𝑡𝑗,2𝑘𝑡𝑗,1||𝑝1,𝑉(𝑝,𝑘)([])𝑢;𝑎,𝑏=sup𝜋𝜎(𝑝,𝑘)(𝑢;𝜋),(3.2) where the supremum is taken over all partitions 𝜋 of the form (3.1). We call the number 𝑉(𝑝,𝑘)(𝑢;[𝑎,𝑏]) the (𝑝,𝑘)-variation of 𝑢 on [𝑎,𝑏]. In case 𝑉(𝑝,𝑘)(𝑢;[𝑎,𝑏])<, we say that the function 𝑢 has bounded (𝑝,𝑘)-variation, and we denote the space of such functions by 𝑅𝑉(𝑝,𝑘)[𝑎,𝑏].

Proposition 3.1. For 𝑘1 and 𝑝(1,), the inclusion 𝑅𝑉(𝑝,𝑘)[]𝑎,𝑏BV𝑘[]𝑎,𝑏(3.3) and the inequality 𝑉𝑘[](𝑢;𝑎,𝑏)(𝑏𝑎)11/𝑝𝑉(𝑝,𝑘)[])(𝑢;𝑎,𝑏1/𝑝(3.4) hold true, where 𝑉𝑘(𝑢;[𝑎,𝑏]) is given by (2.14).

Proof. Let 𝜋 be a partition of [𝑎,𝑏] of the form (3.1). By Hölder’s inequality, we obtain 𝑚𝑗=1||𝑢𝑡𝑗,𝑘+1,,𝑡𝑗,2𝑘𝑡𝑢𝑗,1,,𝑡𝑗,𝑘||||𝑡𝑗,2𝑘𝑡𝑗,1||||𝑡𝑗,2𝑘𝑡𝑗,1||𝑚𝑗=1||𝑢𝑡𝑗,𝑘+1,,𝑡𝑗,2𝑘𝑡𝑢𝑗,1,,𝑡𝑗,𝑘||||𝑡𝑗,2𝑘𝑡𝑗,1||𝑝11/𝑝𝑚𝑗=1||𝑡𝑗,2𝑘𝑡𝑗,1||11/𝑝.(3.5) Passing on both sides of to the supremum over all partitions 𝜋 of the form (3.1), we conclude that (3.4) is true, and so also (3.3), by definition of BV𝑘[𝑎,𝑏].

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

Corollary 3.2. Let 1<𝑝<, 𝑘2, and 𝑢𝑅𝑉(𝑝,𝑘)[𝑎,𝑏]. Then, the derivative 𝑢(𝑘2) belongs to 𝐵𝑉2[𝑎,𝑏]; therefore, the unilateral right and the unilateral left derivatives 𝑢+(𝑘1) and 𝑢(𝑘1) exist on [𝑎,𝑏] and are right and left continuous, respectively.

Now we prove a statement on the existence of the derivative 𝑢(𝑘1)(𝑡0) for all 𝑡0(𝑎,𝑏) in case 𝑢𝑅𝑉(𝑝,𝑘)[𝑎,𝑏].

Proposition 3.3. Let 1<𝑝<, 𝑘2, and 𝑢𝑅𝑉(𝑝,𝑘)[𝑎,𝑏]. Then, the derivative 𝑢(𝑘1)(𝑡) exists for all 𝑡(𝑎,𝑏).

Proof. Suppose that there exists 𝑥0(𝑎,𝑏) such that 𝛼𝑥0=|𝑢+(𝑘1)(𝑥0)𝑢(𝑘1)(𝑥0)|>0. Let 𝜋 be a partition of [𝑎,𝑏] of the form 𝑎=𝑡1,1<𝑡1,2<<𝑡1,𝑘𝑡1,𝑘+1<<𝑡1,2𝑘<𝑡𝑗,1=𝑥0<𝑡𝑗,2<<𝑡𝑗,𝑘<<𝑡𝑗,2𝑘1=𝑥0<𝑡𝑗,2𝑘=𝑥0+<<𝑡𝑚,1𝑡𝑚,2<𝑡𝑚,𝑘𝑡𝑚,𝑘+1<𝑡𝑚,2𝑘=𝑏,(3.6) where satisfies 𝑡0<<𝑚𝑖𝑛𝑗,2𝑘𝑡𝑗,12.𝑗=1,2,,𝑚(3.7)
By definition of the (𝑝,𝑘)-variation of 𝑢, we have then 𝑉(𝑝,𝑘)[]||𝑢𝑡(𝑢;𝑎,𝑏)𝑗,𝑘+1,𝑡𝑗,𝑘+2,,𝑡𝑗,2𝑘2,𝑥0,𝑥0𝑥+𝑢0,𝑡𝑗,2,,𝑡𝑗,𝑘||𝑝||||2𝑝1.(3.8)
From Corollary 3.2 we know that the unilateral right derivative 𝑢+(𝑘1) and the unilateral left derivative 𝑢(𝑘1) are right and left continuous, respectively, on [𝑎,𝑏]. Now, by Corollary 3.2, we further have 𝑉(𝑝,𝑘)[](𝑢;𝑎,𝑏)lim0+lim𝑡𝑗,2𝑥0||𝑢𝑡𝑗,𝑘+1,𝑡𝑗,𝑘+2,,𝑡𝑗,2𝑘2,𝑥0,𝑥0𝑥+𝑢0,𝑡𝑗,2,,𝑡𝑗,𝑘||𝑝||||2𝑝1=𝛼𝑝𝑥02𝑝1((𝑘1)!)𝑝lim0+1𝑝1=,(3.9) contradicting our assumption 𝑢𝑅𝑉(𝑝,𝑘). Consequently, the function 𝑢 has a derivative 𝑢(𝑘1) everywhere on (𝑎,𝑏).

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

Theorem 3.4. Let 1<𝑝< and 𝑘2. Then, 𝑢𝑅𝑉(𝑝,𝑘)[𝑎,𝑏] if and only if 𝑢𝐴(𝑝,𝑘)[𝑎,𝑏]. Moreover, in this case the (𝑝,𝑘)-variation of 𝑢 on [𝑎,𝑏] is given by (1.2).

Proof. Suppose first that 𝑢[𝑎,𝑏] belongs to 𝑅𝑉(𝑝,𝑘)[𝑎,𝑏]. By Proposition 3.3, we know then that the derivative 𝑢(𝑘1) exists on [𝑎,𝑏].
Let 𝜋={𝑎=𝑡0<𝑡1<<𝑡𝑚=𝑏} be a partition of [𝑎,𝑏]. For any subinterval [𝑡𝑗,𝑡𝑗+1], 𝑗=0,1,, 𝑚1, we define a partition 𝜋𝑗 in the form 𝑡𝑗=𝑡𝑗,1<𝑡𝑗,2<<𝑡𝑗,𝑘1<𝑡𝑗,𝑘=𝑡𝑗+<<𝑡𝑗,𝑘+1=𝑡𝑗+1<𝑡𝑗,𝑘+2<<𝑡𝑗,2𝑘=𝑡𝑗+1,(3.10) where 𝑡=min𝑗+1𝑡𝑗2.𝑗=1,2,,𝑚1(3.11)
Using the shortcut 𝑈𝑗,𝑘𝜋𝑗||𝑢𝑡;=𝑗+1,𝑡𝑗,𝑘+2,,𝑡𝑗,2𝑘1𝑡𝑢𝑗,𝑡𝑗,2,,𝑡𝑗,𝑘1,𝑡𝑗||+𝑝||𝑡𝑗+1𝑡𝑗||𝑝1,(3.12) by Corollary 3.2, we have then ||𝑢(𝑘1)𝑡𝑗+1𝑢(𝑘1)𝑡𝑗||𝑝||𝑡𝑗+1𝑡𝑗||𝑝1=lim0+lim𝑡𝑗,𝑘1𝑡𝑗lim𝑡𝑗,𝑘+2𝑡𝑗+1𝑈𝑗,𝑘𝜋𝑗.;(3.13)
Consequently, 1((𝑘1)!)𝑝𝑚𝑗=1||𝑢(𝑘1)𝑡𝑗+1𝑢(𝑘1)𝑡𝑗||𝑝||𝑡𝑗+1𝑡𝑗||𝑝1=lim0+lim𝑡𝑗,𝑘1𝑡𝑗lim𝑡𝑗,𝑘+2𝑡𝑗+1𝑈𝑗,𝑘𝜋𝑗;𝑉(𝑝,𝑘)[](𝑢;𝑎,𝑏).(3.14) Hence, by the Riesz lemma, we see that 𝑢(𝑘1)AC[𝑎,𝑏] and 𝑢(𝑘)𝐿𝑝[𝑎,𝑏], that is, 𝑢𝐴(𝑝,𝑘)[𝑎,𝑏]. Moreover, 𝑏𝑎||𝑢(𝑘)||(𝑡)𝑝((𝑘1)!)𝑝𝑑𝑡𝑉(𝑝,𝑘)[](𝑢;𝑎,𝑏).(3.15)
Conversely, suppose now that 𝑢[𝑎,𝑏] belongs to 𝐴(𝑝,𝑘)[𝑎,𝑏], and let 𝜋 be a partition of [𝑎,𝑏] of the form (3.10). Since 𝑢(𝑘1)AC[𝑎,𝑏] and 𝑢(𝑘)𝐿𝑝[𝑎,𝑏], by Proposition 2.4, we have 𝑚𝑗=1||𝑢𝑡𝑗,𝑘+1,,𝑡𝑗,2𝑘𝑡𝑢𝑗,1,,𝑡𝑗,𝑘||𝑝||𝑡𝑗,2𝑘𝑡𝑗,1||𝑝1=𝑚𝑗=1||𝑢(𝑘1)𝜉𝑗,2𝑘𝑢(𝑘1)𝜉𝑗1,𝑘||𝑝((𝑘1)!)𝑝||𝑡𝑗,2𝑘𝑡𝑗,1||𝑝1=𝑚𝑗=1||||𝜉𝑗𝜉,2𝑘𝑗1,𝑘𝑢(𝑘)||||(𝜎)𝑑𝜎𝑝1((𝑘1)!)𝑝||𝑡𝑗,2𝑘𝑡𝑗,1||𝑝1,(3.16) where 𝜉𝑗,2𝑘conv{𝑡𝑗,𝑘+1,,𝑡𝑗,2𝑘} and 𝜉𝑗1,𝑘conv{𝑡𝑗,1,,𝑡𝑗,𝑘}. By Hölder’s inequality, we get then the estimates 𝑚𝑗=1||||𝜉𝑗,2𝑘𝜉𝑗1,𝑘𝑢(𝑘)||||(𝑡)𝑑𝑡𝑝1((𝑘1)!)𝑝||𝑡𝑗,2𝑘𝑡𝑗,1||𝑝1𝑚𝑗=1𝜉𝑗,2𝑘𝜉𝑗1,𝑘||𝑢(𝑘)||(𝑡)𝑝||𝜉𝑑𝑡𝑗,2𝑘𝜉𝑗1,𝑘||𝑝1((𝑘1)!)𝑝||𝑡𝑗,2𝑘𝑡𝑗,1||𝑝1𝑚𝑗=1𝜉𝑗,2𝑘𝜉𝑗1,𝑘||𝑢(𝑘)||(𝑡)𝑝((𝑘1)!)𝑝𝑑𝑡𝑚𝑗=1𝑡𝑗,2𝑘𝑡𝑗,1||𝑢(𝑘)||(𝑡)𝑝((𝑘1)!)𝑝𝑑𝑡=𝑏𝑎||𝑢(𝑘)||(𝑡)𝑝((𝑘1)!)𝑝𝑑𝑡.(3.17)
This implies that 𝑢𝑅𝑉(𝑝,𝑘)[𝑎,𝑏] and 𝑏𝑎||𝑢(𝑘)||(𝑡)𝑝((𝑘1)!)𝑝𝑑𝑡𝑉(𝑝,𝑘)[](𝑢;𝑎,𝑏).(3.18)
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 𝑅𝑉(𝑝,𝑘)[𝑎,𝑏] included in Theorem 3.4 can be considered as a natural generalization of that given by Riesz [5] for the class 𝐴𝑝[𝑎,𝑏](1<𝑝<), and by Merentes [4] for the class 𝐴(𝑝,2)[𝑎,𝑏].

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

  1. C. Jordan, “Sur la séries de Fourier,” Comptes Rendus de l'Académie des Sciences, vol. 2, pp. 228–230, 1881. View at Google Scholar
  2. Ch. J. De La Vallée Poussin, “Sur la Convergence des formules d'interpolatin entre ordennées equidistantes,” Bulletin de l'Académie Royale des Sciences de Belgique, pp. 314–410, 1908. View at Publisher · View at Google Scholar
  3. M. T. Popoviciu, “Sur quelques propriétés des fonctions d'une variable réelle convexe d'ordre superieure,” Mathematica (Cluj), vol. 8, pp. 1–85, 1934. View at Google Scholar
  4. N. Merentes, “On function of bounded (p,2)-variation,” Collectanea Mathematica, vol. 43, no. 2, pp. 117–123, 1992. View at Google Scholar
  5. F. Riesz, “Untersuchungen über Systeme integrierbarer Funktionen,” Mathematische Annalen, vol. 69, no. 4, pp. 449–497, 1910. View at Publisher · View at Google Scholar
  6. S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics, Izdat. Leningr. Univ., Leningrad, Russia, 1950.
  7. A. M. Russell, “Functions of bounded kth variation,” Proceedings of the London Mathematical Society. Third Series, vol. 26, pp. 547–563, 1973. View at Publisher · View at Google Scholar
  8. A. M. Russell, “Further results on an integral representation of functions of generalised variation,” Bulletin of the Australian Mathematical Society, vol. 18, no. 3, pp. 407–420, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. E. Isaacson and H. B. Keller, Analysis of Numerical Methods, John Wiley & Sons, New York, NY, USA, 1966.