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Azócar, A.; Mejía, O.; Merentes, N.; Rivas, S.

doi:https://doi.org/10.1155/2015/727312

Journal of Function Spaces

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Research Article | Open Access

Volume 2015 | Article ID 727312 | https://doi.org/10.1155/2015/727312

The Spaces of Functions of Two Variables of Bounded -Variation in the Sense of Schramm-Korenblum

A. Azócar,¹O. Mejía,²N. Merentes,²and S. Rivas¹

Academic Editor: Kehe Zhu

Received20 Mar 2015

Accepted10 May 2015

Published07 Jul 2015

Abstract

The purpose of this paper is twofold. Firstly, we introduce the concept of bounded -variation in the sense of Schramm-Korenblum for real functions with domain in a rectangle of . Secondly, we study some properties of these functions and we prove that the space generated by these functions has a structure of Banach algebra.

1. Introduction

During the last decades, several developments, extensions, and generalizations have been considered for the classical concept of the total variation of a function. It is well known that such extensions and generalization play significant role and find many applications in different areas of mathematics. In this paper we introduce a new definition of variation for functions defined on a nonempty rectangle subset of the plane, and we examine the algebra of functions of two variables of bounded generalized variation one obtains from this new definition. Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in a metal plate. The development of the theory of Fourier series in mathematical analysis began in the 19th century and it has been a source of new ideas for analysis during the last two centuries and is likely to be so in years to come. The first exactly proved result was published in Dirichlet’s paper in 1829. That theorem concerns the convergence of Fourier series of piecewise monotonic functions. According to Lakatos [1], functions of bounded variation were discovered by Jordan through a “critical reexamination” of Dirichlet’s famous flawed proof that arbitrary functions can be represented by Fourier series. Jordan [2] proved that if a continuous function has bounded variation, then its Fourier series converges uniformly on a closed bounded set [3]. Jordan gave the characterization of such functions as differences of increasing functions. It is well known that the space of functions of bounded variation on a compact interval is a commutative Banach algebra with respect to pointwise multiplications [4–6]. Functions of bounded variation of one variable are of great interest and usefulness because of their valuable properties. Such properties, particularly with respect to additivity, decomposability into monotone functions, continuity, differentiability, measurability, integrability, and so on, have been much studied. It is largely to the possession of these properties that functions of bounded variation owe their important role in the study of rectifiable curves, Fourier series, Walsh-Fourier series, and other series, Stieltjes integrals, Henstock-Kurzweil integral, and other integrals, and the calculus of variations [7]. Consequently, the study of notions of generalized bounded variation forms an important direction in the field of mathematical analysis [8, 9]. Two well-known generalizations are the functions of bounded -variation and the functions of bounded -variation, due to N. Wiener and L. C. Young, respectively. In 1924 Wiener showed that the Fourier series of function in one variable of finite -variation converges almost everywhere. In 1938 L. C. Young developed an integration theory with respect to functions of finite -variation and showed that the Fourier series of such functions converges everywhere. In 1972 Waterman [10] studied a class of bounded -variation. Combining the notion of bounded -variation with that of bounded -variation, Leindler [11] introduced the class of functions of bounded -variation, and both classes of bounded -variation and bounded -variation are its special cases. In 1980 Shiba [12] introduced the class expanding a fundamental concept of bounded -variation formulated by Waterman.

In 1986, S. K. Kim and J. Kim [13] introduced the notion of functions of bounded -variation on compact interval which is a combination of concepts of bounded -variation and bounded -variation in the sense of Schramm [14]. In [15, 16] Castillo et al. introduce the notion of bounded -variation in the sense of Riesz-Korenblum, which is a combination of the notions of bounded -variation in the sense of Riesz and bounded -variation in the sense of Korenblum. In the year 2014 Guerrero et al. [17] have introduced the space of the functions of two variables of bounded -variation in the sense of Hardy-Vitali-Korenblum and showed that the space is a Banach space.

Soon after Jordan’s work, many mathematicians began to study notions of bounded variation for functions of several variables. There is no uniquely suitable way to extend the notion of variation to function of more than one variable. Proposers of definitions of bounded variation for functions of two variables have been actuated mainly by the desire to single out for attention a class of functions having properties analogous to some particular properties of a function of one variable of bounded variation. Clarkson and Adams [18] study six such generalizations, and Adams and Clarkson [19] mention two more. Two of these definitions are relevant to our purpose. Clarkson and Adams attribute the first to Vitali, Lebesgue, Fréchet, and De la Vallée Poussin and the second to Hardy [20] and Krause [21]. We will refer to them as Vitali variation and Hardy-Krause variation, respectively. Owen [22] provides a very helpful discussion of the concepts of Vitali and Hardy-Krause variation. Another useful reference is Hobson [23]. At the beginning of the past century Hardy [20] generalized the Jordan criterion to the double Fourier series and he proved that if a continuous function of two variables has bounded variation (in the sense of Hardy), then its Fourier series converges uniformly.

Motivated by [13, 17] we introduce for functions of two variables the concept of bounded -variation in the sense of Schramm-Korenblum, which is a suitable combination of the notions of bounded -variation in the sense of Schramm and bounded -variation in the sense of Korenblum for real functions defined on a rectangle of the plane. Our paper is structured as follows. Section 2 provides a review of the notion of Vitali and Hardy-Krause variations for multivariate functions. We recall some notions of variation and introduce for functions of two variables the definitions of bounded -variation in the sense of Schramm-Korenblum. In Section 3 we state and prove our main result: the linear space generated by the class of all bounded -variation functions is a Banach algebra.

2. Preliminaries, Background, and Notations

We begin with some general notation and definitions systematically used throughout the paper.

As usual if and are nonempty sets the symbol denotes the family of functions . We denote by the set of all permutations of set positive integer.

Let be any nonempty interval. For , we define , and if we say that is bounded. We denote by the class of all partitions of . Let and , and the following notations are used frequently:

We will establish the following , , , , , and integer. If , and are partitions of the intervals , , respectively, denoted by

Definition 1. A function is called distortion function if is continuous, increasing, and concave and satisfies , and .

The set of all distortion functions will be denoted by . A distortion function is always subadditive (see [6, Section 2.5 page 170]) in the sense that

Important special cases for choices of are and , (see [24], [25, Section 5], and [26]).

Definition 2 (see [14]). A is a -sequence if is a sequence of increasing convex functions, defined on such that , for all and , and diverges for .

Throughout the paper the double sequence will be a -sequence if for or fixed is a -sequence. It is worthy to recall that the initial works on double sequences can be found in [27, 28].

In what follows we recall different notions of generalized bounded variation.

The notion of variation was introduced by Jordan in 1881 in the one-dimensional case and Vitali and Hardy, which generalized the notion given by Jordan, in 1904–1906 (see [20, 29]). This generalization is for functions of two variables.

Definition 3 (see [2]). The function is of bounded variation ifwhere the supremum is taken over all partitions . We denote by the space of all functions of bounded variation and it is known that is a Banach algebra with respect to the norm , .

Definition 4 (see [20, 29]). Let and be fixed. The Jordan variation of the function is denoted bywhere the supremum is taken over all partitions .
For is fixed, the variation of Jordan of function is defined bywhere the supremum is taken over all partitions .
The variation of in the sense of Hardy-Vitali in the rectangle is defined bywhere the supremum is taken over all partitions and .
The total variation of the function is defined byWe denote by the space of all functions having bounded total variation finite.

The notion of -variation was introduced by Korenblum in 1975 (see [24]) in one-dimensional case and Guerrero et al. in 2015 (see [17]) in two-dimension case.

Definition 5 (see [24]). Let , and the function is of bounded -variation ifwhere the supremum is taken over all partitions . We denote by the space of functions of bounded -variation on .

Definition 6 (see [17]). Let , , and be fixed, and the Jordan -variation of the function is denoted bywhere the supremum is taken over all partitions .
For is fixed, the Jordan -variation of the function is defined bywhere the supremum is taken over all partitions .
The two-dimensional Hardy-Vitali -variation of in the rectangle is defined bywhere the supremum is taken over all partitions and .
The total -variation of the function is defined byWe denote by the space of all functions having bounded -variation total.

The notion of -variation was introduced by Schramm in 1985 (see [14]) in one-dimensional case and Ereú et al. in 2010 (see [30]) in two-dimensional case.

Definition 7 (see [14]). Let be a -sequence and . The function is of bounded -variation ifwhere the supremum is taken over all partitions and . The class of functions with bounded -variation is denoted by and the space generated by this class is denoted by .

Definition 8 (see [30]). Let be a -sequence and let and be fixed, and the -variation in the Schramm sense of the function is defined bywhere the supremum is taken over all partitions and .
For is fixed, the -variation, in the Schramm sense of the function , is defined bywhere the supremum is taken over all partitions and .
The bidimensional variation in the sense of Schramm of the function in the rectangle is defined bywhere the supremum is taken over all partitions and of the intervals , , respectively, and , .
The total -variation of the function is defined byThe class of function with total bounded -variation is denoted by and the space generated by this class is denoted by .

In 1986 S. K. Kim and J. Kim (see [13]) combined the concepts of -variation and -variation introduced by Korenblum and Schramm, respectively, to create the concept -variation in the Schramm-Korenblum sense.

Definition 9 (see [13]). Let be a -sequence, , and . The function is of bounded -variation if where the supremum is taken over all partitions and . The vectorial space generated by this class of functions is denoted by .

3. Main Results

In this section we present the main result of this paper, we generalize the concept of -variation in , presented by S. K. Kim and J. Kim in [13], to the two-dimensional total -variation in in the sense of Schramm-Korenblum, and we prove that the space is a Banach algebra.

Definition 10. Let be a -sequence and let , , and be fixed. The variation of the function in the sense of Schramm-Korenblum of the function is denoted bywhere the supremum is taken over all partitions and .
For is fixed, the variation of Schramm-Korenblum of the function is defined bywhere the supremum is taken over all partitions and .
The two-dimensional variation in the sense of Schramm-Korenblum of in the rectangle is defined bywhere the supremum is taken over all partitions and of the intervals , , respectively, and , .
The total bidimensional -variation in the sense of Schramm-Korenblum of the function is defined by

The class of functions with bounded -variation in the sense of Schramm-Korenblum is denoted by and the space generated by this class is denoted by .

Remark 11. If , , , then the -variation in the sense of Schramm-Korenblum coincides with the Hardy-Vitali-Korenblum variation studied by Guerrero et al. in [17].
If , , where is a Young function, then the Schramm-Korenblum -variation is a combination of the concepts of Wiener -variation and Korenblum-Hardy-Vitali bidimensional variation.
For , , , where is a -sequence in the sense of Waterman, our notion is a combination of the Waterman bidimensional variation introduced by Sahakyan and also studied by Sablim (see [31, 32]) with Korenblum variation [24].

The following comprehensive type results give us interesting properties of the space .

Theorem 12. Let . Then (1) and if and only if ;(2) is convex and symmetry set;(3);(4)if and , then . In particular ;(5) and therefore ;(6)if , then is bounded and

Proof. (1) Suppose that ; then . In particular and , ; thereforeThe “only if” part is immediate.
Parts (2) and (4) follow from Definition 10.
(3) is consequences of (2).
(5) Let and . Consider a partition and let us establish the setSince the functions , , are convex and , then The last sum contains at most terms, because otherwise there would exist at least terms, sowhich is absurd. ThereforesoSimilarlyFinallyso the first inclusion holds.
The other inclusion is a consequence of the subadditive of the function .
(6) Let , and thenOn the other handAs , it follows thatThereforeSimilarlyBy inequalities (36) and (37) and the definition of total bidimensional -variation in the sense of Schramm-Korenblum, it follows that

The following proposition allows us to calculate the total bidimensional -variation in the sense of Schramm-Korenblum of a sum of monotone functions.

Proposition 13. Let be a -sequence, , , and , such that , , where and are monotone functions. Then(a), ;(b), ;(c);(d)

Proof. Since the functions , , are convex then the function , , , is increasing.
(a) Let and and let and be partitions of the intervals , , respectively. ThenSince is monotone and is subadditive, we getSince is monotone and is subadditive thenThereforeReciprocallyThus(b) With a similar reasoning as in part (a) we get(c) If , then, . Therefore .
(d) Is a consequence of Definition 10.

Example 14. Particular cases are obtained when is a function given by the following: (1); then ;(2); then .

Below we present two lemmas that play an important role in what follows.

Lemma 15. Let be a -sequence and . Then the operator , defined by , is convex and the setis convex, symmetric, and absorbent.

Proof. The proof is straightforward; see [33].

The Minkowski functional associated with , , defined byis a seminorm.

Lemma 16. Let Then there exist , such that if and only if .

Proof. If , , then there exist , such that ThenIf , , then . Therefore there exists a sequence , such that ; thenTaking limit as and the result follow. The other part is a consequence of the definition of and the continuity of the functions , , and taking limit as we get desired result.
The “only if” part is consequence of the definition of .

In the next theorem we will prove that the functional , defined byis a norm.

Theorem 17. is a normed space.

Proof. As is a seminorm we only need to show that if then .
If , then and , for all . AccordinglyHence . By part (a) of Theorem 12, As , we obtain .

We will use the next lemma at several places in this work.

Lemma 18. Let , , , and be bounded functions. Then (a), , ;(b), , ;(c), ,

Proof. (a) Consider(b) It follows by reasoning as in part (a).
(c) For , are fixed, we obtain

Theorem 19. is an algebra.

Proof. Let ( and not identically zero) and , a -sequence; then there exist and such thatWe set Now, , are fixed, , , , and . Then by Lemma 18 we havethereforeSimilarlywe getNext, combining (58), (59), and (61) with the fact that the functions , , are increasing and convex and , , we obtaintherefore .

The main result of our paper now reads as follows.

Theorem 20. is a Banach space.

Proof. Let be a Cauchy sequence in ; then, for each , there exists , such that if , thenand by Lemma 16 we getBy the definition of and considering we getDue to , , and (63), we obtainSimilarly, we can consider the definition of and getBy the definition of , , and . ConsiderTherefore is a uniformly Cauchy sequence in , so there exist , such that , as By the continuity of the functions , , and Definition 10 we conclude that , as .
With a similar argument

Theorem 21. The space equipped with the norm is a Banach algebra.

Proof. Let such that they are not identically zero or constant functions andBy Lemma 16, . By Lemma 18SimilarlyAsBy inequalities (72) and (73), Definition 1, the convexity of functions , , and the convexity of operator we haveWe also have by Lemma 16Thus . Using Lemma 16 againClearly, all the conditions in Section of [4] are satisfied. Hence the conditions in Section of [4] ensure that the space is a Banach algebra equipped with the norm and by part of Theorem 12 and Lemma 18, this norm is equivalent to norm . This completes the proof.

Conflict of Interests

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

Acknowledgments

This research has been partially supported by the Central Bank of Venezuela. The authors want to give thanks to the library staff of B.C.V for compiling the references.

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Copyright © 2015 A. Azócar 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.

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