Bivariate functions of bounded variation: Fractal dimension and fractional integral

Abstract

In contrast to the univariate case, several definitions are available for the notion of bounded variation for a bivariate function. This article is an attempt to study the Hausdorff dimension and box dimension of the graph of a continuous function defined on a rectangular region in ${\mathbb{R}}^2$, which is of bounded variation according to some of these approaches. We show also that the Riemann–Liouville fractional integral of a function of bounded variation in the sense of Arzelá is of bounded variation in the same sense. Further, we deduce the Hausdorff dimension and box dimension of the graph of the fractional integral of a bivariate continuous function of bounded variation in the sense of Arzelá.

Introduction

This paper is primarily concerned with the concept of bounded variation of a bivariate function. The notion of bounded variation was originally introduced for a real-valued function on a closed bounded interval in $\mathbb{R}$ by Jordan [13]. The concept of bounded variation stimulated interest because of its properties such as additivity, decomposability into monotone functions, continuity, differentiability, measurability and integrability. The functions of bounded variation, for instance, play a major role in the study of rectifiable curves, Fourier series, integrals and calculus of variations.

The motivation for the current work is multifold. The first is the theory of bivariate functions of bounded variation, which enjoy interesting connections with various branches of pure and applied mathematics. There is no unique suitable way to extend the notion of variation to a function of more than one variable. Various approaches to the notion of bounded variation of a multivariate function target to identify a class of functions having similar properties as that of a univariate function of bounded variation. Of the several approaches to the concept of bounded variation for functions of several variables, popular versions are attributed to Vitali, Hardy, Arzelá, Pierpont, Fréchet, Tonelli and Hahn. The reader may refer [1], [2], [9] for a comprehensive collection of these seven variants of bounded variation. In fact, new definitions and approaches continue to be introduced for various applications. For more recent generalizations for the concept of total variation of a function, the interested reader may consult [3], [5], [6], [7], [8], [11] and references quoted therein. For applications of multivariate functions of bounded variation in the Koksma–Hlawka type inequalities in equilibrium theory, the reader may refer [4], [10].

Among establishing various properties of a function of bounded variation, calculation of fractal dimension of its graph has gained interest in fractal geometry and related fields. In fractal approximation theory, the Hausdorff dimension and box dimension constitute important quantifiers that need to agree between the constructed approximants and the object being approximated. For definitions and basic results on various approaches to the notion of fractal dimension, the reader is referred to the popular textbook by Falconer [12]. Using the fact that a univariate function of bounded variation can have at most a countable number of discontinuous points and some basic properties of the Hausdorff dimension, it is easy to prove that the Hausdorff dimension of the graph of a univariate function of bounded variation on $[a,b]$ is $1$, see, for instance, [12]. Supplementing this, recently, Liang proved an elementary and elegant result that the box dimension of the graph of a univariate continuous function of bounded variation is $1$ (Theorem 1.3, [14]). This result acts as the second motivating influence for our work herein. To be precise, the aforementioned theorem in reference [14] stimulated to ask if an analogous result for a bivariate function of bounded variation exists. Section 3 seeks to show that this is indeed the case, in fact with a suitable interpretation for the notion of bounded variation. For instance, among others, we prove:

Theorem 1.1

If $f:[a,b]\times [c,d]\to \mathbb{R}$ is continuous and of bounded variation in the sense of Hahn, then the Hausdorff dimension and box dimension of its graph is $2$.

As a prelude to this, we need a bivariate analogue of a well-known proposition (Proposition 11.1, [12]), which is applied to find the bounds for the box dimension of the graph of a univariate continuous function. Although this is a fundamental and natural extension, we did not find it explicitly anywhere in the literature, for which reason we record it in Section 3. Let us note that while univariate functions of bounded variation are relatively easy to deal with, the multivariate theory is intricate with roots in geometric measure theory. However, our exposition has a different goal, that is, to apply some elementary techniques to study the dimension of the graph of a bivariate function of bounded variation.

Fractional calculus, which can be broadly interpreted as the theory of derivatives and integrals of fractional (non-integer) order and their diverse applications, is an older subject dating back nearly 300 years. The literature relevant to fractional calculus is substantial; for a selection, the reader can refer to an encyclopedic book [19]. Perhaps due mostly to linguistic reasons, there have been efforts to relate the two apparently diverse areas — fractional calculus and fractal geometry. Apart from the linguistic reason, research studies to connect fractional calculus with fractals were motivated by the need for physical and geometric interpretations of the fractional order integration and differentiation [17], [18]. In this regard, in [14] it has been deduced that the box dimension of the graph of the Riemann–Liouville fractional integral of a continuous function of bounded variation is $1$. Motivated by this, the last section of the current article establishes the Hausdorff dimension and box dimension of the graph of the mixed Riemann–Liouville fractional integral of a bivariate continuous function of bounded variation in the sense of Arzelá.