Bounded variation of functions defined on a convex and compact set in the plane

Abstract

In this paper, the variation of functions has been defined, whose domain is a convex and compact set in the plane. Furthermore, in addition to presenting properties that satisfy this variation, the vector space formed by functions with finite variation is studied, demonstrating that it is a Banach space and its elements can be expressed as the difference of non-decreasing functions.

2020 Mathematics Subject Classification:

• 26A45
• 26B30

Keywords and Phrases:

• Bounded variation
• compact set

1. Introduction

Even though we have the definition of the integral of functions defined over a rectangle in the plane, to achieve flexibility when studying some natural phenomena, it is necessary to generalize this definition to more general regions, since many of them, perhaps most of them, can be developed in more irregular regions. And in the same line, although it is true that the study of functions of bounded variation arises as a necessary tool in the study of Fourier series, they have acquired importance from the theoretical point of view. Thus, this space of functions has been treated by several authors. First, the initial definition of functions of a real variable defined on a compact interval has now been extended to more general sets, among them: open sets, compact sets, etc. It has even been defined for functions whose values are in metric spaces or Banach spaces. Subsequently, interest arose in working with this type of function defined in two or more variables, see (Ambrosio, 1990; Bracamonte, Giménez, & Merente, 2013; Chistyakov & Tretyachenko, 2010). However, there is interest in studying some types of integral equations, such as those of the Fredholm and/or Volterra type (see Banaś & Dronka, 2000; Ereú, Marchan, Pérez, & Rojas, 2023). However, we are interested in obtaining solutions in the space of functions of bounded variation of nonlinear integral equations such as $\varphi (x)={\int }_0^{x}K(x,y,\varphi (y))dy+f(x)$ where f is a bounded variation function, for which it is essential that the functions that are involved in them are defined in more general sets. This new definition allows the study, among other things, of equations of this type. This is part of the motivation of this study, in addition to the interest of space itself, which is still of interest to many researchers (Bugajewski & Gulgowski, 2020; Ereú et al., 2023; Giménez, Merentes, & Vivas, 2014). We begin thus, with the definition of variation of a function defined on a compact and convex set, a relatively simple set in the plane.

2. Preliminaries

We begin by presenting the most frequently know definition of variation of a function with a rectangle as domain.

Let $\mathbf{R}=[a,b]\times [c,d]$ be a rectangle in ${\mathbb{R}}^2,$ we assume that $a<b$ and $c<d.$ As in (Clarkson & Adams, 1933), the term net we shall, unless otherwise specified, mean a set of parallels to the axes $$\begin{array}{cc}a\hfill & =x_0<x_1<\cdots <x_n=b\hfill \\ c=\hfill & y_0<y_1<\cdots <y_m=d.\hfill \end{array}$$

In this case we will say that $\xi ={\left\{x_i\right\}}_{i=0}^n$ and $\eta ={\left\{y_j\right\}}_{j=0}^m$ are partitions of the intervals [a, b] and [c, d] respectively, which we will denote by $\xi \in \Lambda ([a,b])$ and $\eta \in \Lambda ([c,d]).$ Furthermore the net $\xi \times \eta$ will state that $\xi \times \eta \in \Lambda (\mathbf{R}).$

Each of the smaller rectangles $I_{ij}=[x_{i-1},x_i]\times [y_{j-1},y_j]$ into which $\mathbf{R}$ is partitioned by a net will be called a cell.

If $f:\mathbf{R}\to \mathbb{R},$ the authors also define the today called Vitali difference for $I_{ij}=[x_{i-1},x_i]\times [y_{j-1},y_j]$ by (2.1) $$m{d}_2(f,I_{ij}):=f(x_{i-1},y_{j-1})-f(x_i,y_{j-1})+f(x_i,y_j)-f(x_{i-1},y_j).$$(2.1)

Note that this difference involves all four vertices of the cell $I_{ij}=[x_{i-1},x_i]\times [y_{j-1},y_j],$ therefore it is required that the function f is defined in $\mathbf{R}.$

Let $\xi ={\left\{x_i\right\}}_{i=0}^n$ be a partition of [a, b] and $\eta ={\left\{y_j\right\}}_{j=0}^m$ be a partition of [c, d]. Is clear that, $\xi \times \eta$ is a net for $\mathbf{R}.$ Under the given conditions the following variations are defined $$\begin{array}{cc}{V}_{10}(f,\mathbf{R},\xi )\hfill & :=\sum _{i=1}^n\left|f\left(x_i,c\right)-f\left(x_{i-1},c\right)\right|,\hfill \\ V_{01}(f,\mathbf{R},\eta )\hfill & :=\sum _{j=1}^m\left|f\left(a,y_j\right)-f\left(a,y_{j-1}\right)\right|,\hfill \\ V_{11}(f,\mathbf{R},\xi \times \eta )\hfill & :=\sum _{i=1}^n\sum _{j=1}^m\left|m{d}_2(f,I_{ij})\right|,\hfill \end{array}$$ and $$\begin{array}{cc}{V}_{10}(f,\mathbf{R})\hfill & :=\underset{\xi \in \Lambda ([a,b])}{\sup }{V}_{10}(f,\mathbf{R},\xi )\hfill \\ V_{01}(f,\mathbf{R})\hfill & :=\underset{\eta \in \Lambda ([c,d])}{\sup }{V}_{01}(f,\mathbf{R},\eta )\hfill \\ V_{11}(f,\mathbf{R})\hfill & :=\underset{\begin{array}{c}\xi \in \Lambda ([a,b])\\ \eta \in \Lambda ([c,d])\end{array}}{\sup }{V}_{11}(f,\mathbf{R},\xi \times \eta ),\hfill \end{array}$$ where the supremum is being taken over all $\xi ,\eta$ of partitions of [a, b] and [c, d] respectively.

The total variation of $f:\mathbf{R}\to \mathbb{R}$ is defined by $$TV(f,\mathbf{R}):=V_{10}(f,\mathbf{R})+V_{01}(f,\mathbf{R})+V_{11}(f,\mathbf{R}).$$

In (Chistyakov & Tretyachenko, 2010) the space of functions of finite total variation is defined (in sense of Vitali, Hardy and Krause) as, $$BV(\mathbf{R})=\{f:[a,b]\times [c,d]\to \mathbb{R}\mid TV(f,\mathbf{R})<\infty \}.$$

In addition, several properties are demonstrated, among which are

1. f is bounded,

2. Is $f\in BV(\mathbf{R}),$ and $(x_1,y_1),(x_2,y_2)\in \mathbf{R}$ with $x_1<x_2$ and $y_1<y_2$ then (2.2) $$\begin{array}{c}\begin{array}{cc}|f(x_1,y_1)-f(x_2,y_2)|\hfill & \le |f(x_1,y_1)-f(x_1,y_2)|+|f(x_1,y_1)-f(x_2,y_1)|\hfill \end{array}\\ +|f(x_1,y_1)-f(x_2,y_1)+f(x_2,y_2)-f(x_1,y_2)|.\end{array}$$(2.2)

3. if a sequence of maps $\left\{f_j\right\}$ from $\mathbf{R}$ into $\mathbb{R}$ converges pointwise on $\mathbf{R}$ to a map $f:\mathbf{R}\to \mathbb{R},$ then (2.3) $$TV(f,\mathbf{R})\le \underset{j\to \infty }{\lim \inf }TV(f_j,\mathbf{R})$$(2.3)

4. In (Idczak, 1994), an interval function associated with f is defined as follows $$F_f(P)=f(x_2,y_2)-f(x_2,y_1)-f(x_2,y_2)+f(x_1,y_1)$$ where $P=[x_1,x_2]\times [y_1,y_2]\subseteq \mathbb{R}.$Can be observed that the definition coincides with the one given for $m{d}_2(f,I_{ij}),$ this function is additive and has bounded variation when $$\sup \left\{\sum _{i=1}^n|F_f(P_i)|:\underset{i=1}{\overset{n}{\cup }}{P}_i=\mathbf{R},n\ge 1\right\}<\infty$$ where $P_i,$ $i=1,\dots ,n$ are rectangles such that $\mathit{Int}(P_i)\cap \mathit{Int}(P_j)=\varnothing$ (disjoint inner sets).In other words, this function has bounded variation if $V_{11}(f,\mathbf{R})$ is finite.Furthermore, in (Łojasiewicz, 1988) it has been proven that if this function has bounded variation, it can be decomposed (Jordan Canonical Decomposition) as the difference of two additive, non-negative functions defined for $P\subseteq \mathbf{R}.$

5. $BV(\mathbf{R})$ is a Banach algebra with respect to the usual pointwise operations and norm $$\Vert f{\Vert }_{{}_{BV(\mathbf{R})}}:=|f(a,c)|+TV(f,\mathbf{R}).$$

6. The following inequality holds: $$\begin{array}{cc}\Vert f\cdot g{\Vert }_{{}_{BV(\mathbf{R})}}\le 4\Vert f{\Vert }_{{}_{BV(\mathbf{R})}}\Vert g{\Vert }_{{}_{BV(\mathbf{R})}},\hfill & f,g\in BV(\mathbf{R}).\hfill \end{array}$$

3. Variation of a function defined on more general sets

Our objective in this section is to define the variation of functions whose domain is a more general set $\sigma .$ For which we have restricted ourselves to compact and convex sets.

Let $\sigma$ be a compact and convex set in the plane, we will restrict ourselves to rectangle $\mathbf{R}=[a,b]\times [c,d],$ which is the smallest rectangle that contains $\sigma .$ Note also that if we consider a finite number of points in $\sigma ,$ they determine partitions $\xi$ and $\eta$ in the intervals [a, b] and [c, d] respectively.

When we are working on a rectangle in the plane the total variation is found by adding three variations, the first two corresponding to a variation of only one variable, but just on two of the edges of the rectangle where the variation of the function is being studied. In our case, to replicate this we make some preliminary considerations, let’s see.

Evidently, if $\sigma$ is not a rectangle, then there must exist elements in $\mathbf{R}$ that are not in $\sigma .$ Besides, given a point $(x_0,y_0)$ in $\mathbf{R}$ (may not be in $\sigma$), obviously the lines $x=x_0$ and $y=y_0$ have points in common with $\sigma ,$ otherwise it would imply that $\mathbf{R}$ is not the smallest rectangle containing $\sigma .$

Taking this into account, for each point $x\in [a,b]$ we define (3.1) $$\beta (x):=\inf \{y\in [c,d]:(x,y)\in \sigma \}\text{and}B(x):=\sup \{y\in [c,d]:(x,y)\in \sigma \}.$$(3.1)

Example 1.

If $\sigma$ is the polygon of vertices $(2,0),(3,1),(2,2)$ and $(1,1),$ it is true that $\mathbf{R}=[1,3]\times [0,2]$ will be and consequently for the point $\frac{3}{2}\in [1,3],$ so we have that $$\beta \left(\frac{3}{2}\right)=\frac{1}{2}\text{and}B\left(\frac{3}{2}\right)=\frac{3}{2}.$$

In fact, for any x in [1, 2] one has that $$\beta \left(x\right)=-x+2\text{and}B\left(x\right)=x,$$ while if x is in [2, 3] it follows that $$\beta \left(x\right)=x-2\text{and}B\left(x\right)=4-x.$$

Similarly, for each y in [c, d] we define (3.2) $$\alpha (y):=\inf \{x\in [a,b]:(x,y)\in \sigma \}\text{and}A(y):=\sup \{x\in [a,b]:(x,y)\in \sigma \}.$$(3.2)

In the above example, for any $y\in [0,1]$ we have that $$\alpha \left(y\right)=2-y\text{and}A\left(y\right)=y+2.$$ while, if $y\in [1,2]$ is true that $$\alpha \left(y\right)=y\text{and}A\left(y\right)=4-y.$$

By defining the variation of a function whose domain is $\sigma$

With these resources, we can now introduce the variation of a function defined over $\sigma .$

Definition 3.1.

Let $\xi ={\left\{x_i\right\}}_{i=0}^n$ be a partition of [a, b], we define $$\begin{array}{cc}{V}_{10}(f,\sigma )\hfill & :=\underset{\xi \in \Lambda ([a,b])}{\sup }{V}_{10}(f,\sigma ,\xi ),\hfill \end{array}$$ where $$\begin{array}{cc}{V}_{10}(f,\sigma ,\xi )\hfill & :=\sum _{i=1}^n\left|f\left(x_i,\beta (x_i)\right)-f\left(x_{i-1},\beta (x_{i-1})\right)\right|.\hfill \end{array}$$

Note that $V_{10}(f,\sigma )$ describes the behavior of f at the lower boundary of $\sigma .$

Example 2.

Consider the function $f(x,y)=x+y$ defined on the triangle with vertices at the points $(0,1),(1,0)$ and $(2,1).$ In this case $\mathbf{R}$ is the rectangle with vertices at $(0,0),(2,0),(2,1)$ and $(0,1),$ that is $\mathbf{R}=[0,2]\times [0,1].$ Note that for any partition $\xi \in \Lambda ([0,2]),$ including 1, one will have that $$(x_i,\beta (x_i))=\{\begin{array}{ccc}(x_i,1-x_i)& \text{if}& 0\le x_i\le 1\\ (x_i,x_i-1)& \text{if}& 0\le x_i\le 2\end{array}$$ $$\begin{array}{cc}{V}_{10}(f,\sigma ,\xi )=\sum _{i=1}^k\left|f(x_i,1-x_i)-f(x_{x_i-1},1-x_{i-1})\right|+\sum _{i=k}^n\left|f(x_i,x_i-1)-f(x_{i-1},x_{i-1}-1)\right|\hfill & \hfill \\ =\sum _{i=1}^k\left|1-1\right|+\sum _{i=k}^n\left|2x_i-1-(2x_{i-1}-1)\right|\hfill & \hfill \\ =\sum _{i=k}^n\left|2x_i-2x_{i-1}\right|=2(2-1)=2.\hfill & \hfill \end{array}$$

Since this is true for any partition of [0, 2], it follows that $$V_{10}(f,\sigma )=2.$$

We now proceed to define the variation $V_{01}(f,\sigma ).$ For which, we begin by noting that there exists a $y\in [c,d]$ such that $\alpha (y)=a.$ Therefore, we can define (3.3) $$e=\inf \{y\in [c,d]:\alpha (y)=a\}.$$(3.3)

In fact, this infimum is a minimum. Consequently, we can now define $V_{01}(f,\sigma )$ in the following way.

Definition 3.2.

Let $\eta ={\left\{y_j\right\}}_{j=0}^m$ be a partition of [e, d]; then $V_{01}(f,\sigma )$ is define by $$\begin{array}{cc}{V}_{01}(f,\sigma )\hfill & :=\underset{\eta \in \Lambda ([e,d])}{\sup }{V}_{01}(f,\sigma ,\eta ),\hfill \end{array}$$ where $$\begin{array}{cc}{V}_{01}(f,\sigma ,\eta )\hfill & :=\sum _{i=1}^m\left|f\left(\alpha (y_i),y_i\right)-f\left(\alpha (y_{i-1}),y_{i-1}\right)\right|.\hfill \end{array}$$

Note that $V_{01}(f,\sigma )$ indicates the variation of f over the left boundary of $\sigma ,$ which does not overlap with the lower boundary.

Remark 3.3.

• Note that in the case where $\sigma$ is a rectangle, $e=c$ and indeed $V_{10}(f,\sigma )$ and $V_{01}(f,\sigma )$ are effectively a generalization of the variations $V_{10}(f,R)$ and $V_{01}(f,R)$ that are defined over a rectangle.

• If $e=d,$ $V_{01}(f,\sigma )$ is defined as zero.

Example 3.

Considering sigma and f as in the previous example, in this case, $e=1,$ consequently we have that $$V_{01}(f,\sigma )=0.$$

A generalization of Vitali difference

Therefore, it is necessary to generalize the definition of Vitali difference given in (2.1)(2.1) $$m{d}_2(f,I_{ij}):=f(x_{i-1},y_{j-1})-f(x_i,y_{j-1})+f(x_i,y_j)-f(x_{i-1},y_j).$$(2.1) . With this goal in mind, we define an extension of the function f to $\mathbf{R}$ by $$f_v(x,y):=\{\begin{array}{ccc}f(x,y)& \text{if}& (x,y)\in \sigma \\ f(x,\beta (x))& \text{if}& (x,y)\in \mathbf{R}-\sigma \text{and} y<\beta (x)\\ f(x,B(x))& \text{if}& (x,y)\in \mathbf{R}-\sigma \text{and} y>B(x).\end{array}$$

Then the type Vitali difference of f on the rectangle $I_{ij}\subseteq [a,b]\times [c,d],$ is defined by, $$m{d}_2(f,I_{ij}):=m{d}_2(f_v,I_{ij}).$$

Note that if $I_{ij}\subseteq \sigma$ then $$m{d}_2(f,I_{ij})=f(x_{i-1},y_{j-1})-f(x_i,y_{j-1})+f(x_i,y_j)-f(x_{i-1},y_j).$$

Now we can, emulating the definition of bounded variation for functions whose domain is a rectangle, present the corresponding definition for functions defined on a compact and convex set $\sigma .$ $$\begin{array}{cc}{V}_{11}(f,\sigma )\hfill & :=\underset{\begin{array}{c}\xi \in \Lambda ([a,b])\\ \eta \in \Lambda ([c,d])\end{array}}{\sup }{V}_{11}(f,\sigma ,\xi \times \eta ),\hfill \end{array}$$ where $$\begin{array}{cc}{V}_{11}(f,\sigma ,\xi \times \eta )\hfill & :=\sum _{i=1}^n\sum _{j=1}^m\left|m{d}_2(f,I_{ij})\right|,\hfill \end{array}$$ and (3.4) $$\begin{array}{cc}TV(f,\sigma )\hfill & :=V_{10}(f,\sigma )+V_{01}(f,\sigma )+V_{11}(f,\sigma ).\hfill \end{array}$$(3.4)

We also present the set of functions of finite total variation (in the sense of Vitali – Hardy and Krause) as $$BV(\sigma ):=\{f:\sigma \to \mathbb{R}\mid TV(f,\sigma )<\infty \}.$$

It is clear that the particular case in which $\sigma$ is a rectangle already fulfills the properties stated and studied above. So we now turn to study what properties are preserved when sigma is a more general set.

Properties

It is expected that this newly defined variation satisfies analogous properties to those satisfied by the variation in functions defined on a rectangle R. Let’s take a look at some of them.

• (P1) If $f,g\in BV(\sigma ),$ $\lambda ,\gamma \in \mathbb{R},$ $\xi ={\left\{x_i\right\}}_{i=0}^n$ and $\eta ={\left\{y_i\right\}}_{i=0}^m$ are partitions of [a, b] and [e, d] (e as it has been defined in (3.3)(3.3) $$e=\inf \{y\in [c,d]:\alpha (y)=a\}.$$(3.3) ) respectively, then for each $1\le i\le n$ we will have, $$\begin{array}{c}\left|(\lambda f+\gamma g)\left(x_i,\beta (x_i)\right)-(\lambda f+\gamma g)\left(x_{i-1},\beta (x_{i-1})\right)\right|\\ =\left|\lambda f\left(x_i,\beta (x_i)\right)+\gamma g\left(x_i,\beta (x_i)\right)-\lambda f\left(x_{i-1},\beta (x_{i-1})\right)-\gamma g\left(x_{i-1},\beta (x_{i-1})\right)\right|\\ \le \left|\lambda \right|\left|f\left(x_i,\beta (x_i)\right)-f\left(x_{i-1},\beta (x_{i-1})\right)\right|+\left|\gamma \right|\left|g\left(x_i,\beta (x_i)\right)-g\left(x_{i-1},\beta (x_{i-1})\right)\right|.\end{array}$$

So, (3.5) $$V_{10}(\lambda f+\gamma g,\sigma )\le \left|\lambda \right|V_{10}(f,\sigma )+\left|\gamma \right|V_{10}(g,\sigma ).$$(3.5)

Similarly, $$\begin{array}{c}\left|(\lambda f+\gamma g)\left(\alpha (y_i),y_i\right)-(\lambda f+\gamma g)\left(\alpha (y_{i-1}),y_{i-1}\right)\right|\\ \le \left|\lambda \right|\left|f\left(\alpha (y_i),y_i\right)-f\left(\alpha (y_{i-1}),y_{i-1}\right)\right|+\left|\gamma \right|\left|g\left(\alpha (y_i),y_i\right)-g\left(\alpha (y_{i-1}),y_{i-1}\right)\right|,\end{array}$$ which guarantees that (3.6) $$V_{01}(\lambda f+\gamma g,\sigma )\le \left|\lambda \right|V_{01}(f,\sigma )+\left|\gamma \right|V_{01}(g,\sigma ).$$(3.6)

Given that $f_v$ and $g_v$ are defined over the rectangle $\mathbf{R},$ then they satisfy the inequality $$V_{11}(\lambda f_v+\gamma g_v,\mathbf{R})\le \left|\lambda \right|V_{11}(f_v,\mathbf{R})+\left|\gamma \right|V_{11}(g_v,\mathbf{R}),$$ and by its definition, it guarantees us that (3.7) $$V_{11}(\lambda f+\gamma g,\sigma )\le \left|\lambda \right|V_{11}(f,\sigma )+\left|\gamma \right|V_{11}(g,\sigma ).$$(3.7)

Thus, from inequalities (3.5)(3.5) $$V_{10}(\lambda f+\gamma g,\sigma )\le \left|\lambda \right|V_{10}(f,\sigma )+\left|\gamma \right|V_{10}(g,\sigma ).$$(3.5) , (3.6)(3.6) $$V_{01}(\lambda f+\gamma g,\sigma )\le \left|\lambda \right|V_{01}(f,\sigma )+\left|\gamma \right|V_{01}(g,\sigma ).$$(3.6) and (3.7)(3.7) $$V_{11}(\lambda f+\gamma g,\sigma )\le \left|\lambda \right|V_{11}(f,\sigma )+\left|\gamma \right|V_{11}(g,\sigma ).$$(3.7) , is obtained that $\lambda f+\gamma g\in BV(\sigma )$ and $$\begin{array}{cc}TV(\lambda f+\gamma g,\sigma )\hfill & \le \left|\lambda \right|TV(f,\sigma )+\left|\gamma \right|TV(g,\sigma ).\hfill \end{array}$$

• (P2) Since the null function is in $BV(\sigma )$ and property (P1) is satisfied we obtain that, $BV(\sigma ),$ with the operations of addition and multiplication by a scalar, is a real vector space.

• (P3) If $\xi ={\left\{x_i\right\}}_{i=0}^n$ is a partition of [a, b] and t is a point in $(a,b)$ that is not in the partition, then $t\in [x_{k-1},x_k]$ for some $1\le k\le n$ and consequently using the triangular inequality in the $k-$th summand of

$$\sum _{i=1}^n\left|f\left(x_i,\beta (x_i)\right)-f\left(x_{i-1},\beta (x_{i-1})\right)\right|$$

we obtain $$V_{10}(f,\sigma ,\xi )\le V_{10}(f,\sigma ,\xi \cup \{t\}).$$

Similarly, it can be verified that $$V_{01}(f,\sigma ,\eta )\le V_{01}(f,\sigma ,\eta \cup \{s\}).$$

Using the properties of Vitali difference for functions defined on a rectangle, we obtain that $$V_{11}(f,\sigma ,\xi \times \eta )\le V_{11}(f,\sigma ,\xi \times \eta \cup \{(t,s)\}).$$

The importance of this property is that it establishes a monotonicity of variation when we add points to the partitions. This allows us to treat the supremum as a limit when necessary.

• (P4) If $f\in BV(\sigma )$ then f is a bounded function. Note that $$\sum _{i=1}^n\left|f_v\left(a,y_i\right)-f_v\left(a,y_{i-1}\right)\right|=0$$

except possibly if the interval $[\beta (a),B(a)]$ contains more than one point. In which case we will have that $$\begin{array}{cc}{V}_{10}(f_v,\mathbf{R})\hfill & =V_{10}(f,\sigma )\hfill \\ V_{01}(f_v,\mathbf{R})\hfill & \le V_{01}(f,\sigma ).\hfill \end{array}$$

From this we can state that if $f\in BV(\sigma )$ then $f_v\in BV(\mathbf{R}).$A consequence of this is that $f_v$ is a bounded function, and in fact, if $(x,y)\in \sigma$ then $$\begin{array}{cc}|f(x,y)|=|f_v(x,y)|\hfill & \hfill \\ \le V_{11}(f_v,\mathbf{R})+V_{10}(f_v,\mathbf{R})+V_{01}(f_v,\mathbf{R})+|f_v(a,c)|\hfill & \hfill \\ \le V_{11}(f,\sigma )+V_{10}(f,\sigma )+V_{01}(f,\sigma )+|f_v(a,c)|\hfill & \hfill \\ =TV(f,\sigma )+|f(a,\beta (a))|.\hfill & \hfill \end{array}$$

It is crucial to understand the ”magnitude” of this novel class of functions, as it would not significantly contribute if it solely comprised constant functions. Theorem 3.7 will provide the justification for investigating and studying this distinct class of functions.

Lemma 3.4.

Let $f:\sigma \to \mathbb{R}$ be a Lipschitz continuous function, with Lipschitz constant L, then $$V_{10}(f,\sigma )\le L\delta (\sigma )$$ where $\delta (\sigma )$ denotes the diameter of $\sigma .$

Proof.

If $f:\sigma \to \mathbb{R}$ be a Lipschitz continuous function, with Lipschitz constant L, $\xi ={\left\{x_i\right\}}_{i=0}^n$ is a partition of [a, b], then for $1\le i\le n$ we have $$\begin{array}{cc}\left|f\left(x_i,\beta (x_i)\right)-f\left(x_{i-1},\beta (x_{i-1})\right)\right|\hfill & \le L\Vert \left(x_i,\beta (x_i)\right)-\left(x_{i-1},\beta (x_{i-1})\right){\Vert }_2\hfill \\ =L\Vert \left(x_i-x_{i-1},\beta (x_i)-\beta (x_{i-1})\right){\Vert }_2\hfill & \hfill \\ \le L\delta (\sigma ).\hfill & \hfill \end{array}$$

Consequently, we obtain that $$\begin{array}{cc}{V}_{10}(f,\sigma ,\xi )\hfill & \le L\delta (\sigma ).\hfill \end{array}$$

Similarly, the following lemma can be demonstrated.

Lemma 3.5.

Let $f:\sigma \to \mathbb{R}$ be a Lipschitz continuous function, with Lipschitz constant L, then $$V_{01}(f,\sigma )\le L\delta (\sigma )$$ where $\delta (\sigma )$ denotes the diameter of $\sigma .$

What happens with the variation of both variables? let’s see.

Lemma 3.6

Let $f:\sigma \to \mathbb{R}$ be a Lipschitz continuous function, with Lipschitz constant L then $V_{11}(f,\sigma )\le 2L\min \{b-a,d-c\}.$

Proof.

If $f:\sigma \to \mathbb{R}$ be a Lipschitz continuous function, with Lipschitz constant L, $\xi ={\left\{x_i\right\}}_{i=0}^n$ and $\eta ={\left\{y_j\right\}}_{j=0}^m$ partitions of [a, b] and [c, d] respectively, then $$\begin{array}{cc}m{d}_2(f,I_{ij})=|f_v(x_{i-1},y_{j-1})-f_v(x_i,y_{j-1})+f_v(x_i,y_j)-f_v(x_{i-1},y_j)|\hfill & \hfill \\ \le |f_v(x_{i-1},y_{j-1})-f_v(x_i,y_{j-1})|+|f_v(x_i,y_j)-f_v(x_{i-1},y_j)|\hfill & \hfill \\ \le 2L|x_i-x_{i-1}|\hfill & \hfill \end{array}$$ and $$\begin{array}{cc}m{d}_2(f,I_{ij})=|f_v(x_{i-1},y_{j-1})-f_v(x_i,y_{j-1})+f_v(x_i,y_j)-f_v(x_{i-1},y_j)|\hfill & \hfill \\ \le |f_v(x_{i-1},y_{j-1})-f_v(x_{i-1},y_j)|+|f_v(x_i,y_j)-f_v(x_i,y_{j-1})|\hfill & \hfill \\ \le 2L|y_j-y_{j-1}|.\hfill & \hfill \end{array}$$

From which we obtain that $$\begin{array}{cc}{V}_{11}(f,\sigma ,\xi \times \eta )\hfill & =\sum _{i=1}^n\sum _{j=1}^m\left|m{d}_2(f,I_{ij})\right|\le 2L\min \{b-a,d-c\}.\hfill \end{array}$$

So $$\begin{array}{cc}{V}_{11}(f,\sigma )\hfill & \le 2L\min \{b-a,d-c\}.\hfill \end{array}$$

Note that if there must exist points $(x_0,c),(x_1,d)\in \mathbf{R}$ such that $f(x_0)=c$ and $f(x_1)=d$ in which case $d-c\le \delta (\sigma ).$ And with the same reasoning it follows that $b-1\le \delta (\sigma ).$

As a consequence the following theorem can be stated

Theorem 3.7.

Let $f:\sigma \to \mathbb{R}$ be a Lipschitz continuous function, with Lipschitz constant L, then $f\in BV(\sigma )$ and $$TV(f,\sigma )\le 4L\delta (\sigma ),$$ where $\delta (\sigma )$ denotes the diameter of $\sigma .$

4. The space $BV(\sigma )$

As a consequence of (P1) we have that $TV(\cdot ,\sigma )$ is a semi-norm. Then is worth asking whether a norm can be defined in this set of functions. For which it will be essential to consider the following property.

Lemma 4.1.

$TV(f,\sigma )=0$ if and only if f is constant.

Proof.

It is clear that if f is constant then $TV(f,\sigma )=0.$ On the other hand, given that $f\in BV(\sigma )$ then $f_v\in BV(\mathbf{R})$ and $TV(f_v,\mathbf{R})=0,$ in this case $f_v$ is constant, therefore so is f. ▪

Lemma 4.2.

$\Vert f{\Vert }_{BV(\sigma )}:=|f(a,\beta (a))|+TV(f,\sigma )$ defines a norm for $BV(\sigma ).$

The proof is an immediate consequence of property (P1) and the previous lemma.

Theorem 4.3

$BV(\sigma )$ is a Banach space.

Proof.

Let ${(f_n)}_n$ be a Cauchy sequence in $BV(\sigma ),$ then for each $(x,y)\in \sigma$ and $ϵ>0.$ Since ${(f_n)}_n$ is a Cauchy sequence, there exists $N>0$ such that $$\Vert f_n-f_m{\Vert }_{BV(\sigma )}<ϵ\text{for each}n,m\ge N.$$

Then, if $n,m\ge N$ $$\begin{array}{cc}|f_n(x,y)-f_m(x,y)|\hfill & \le TV(f_n-f_m,\sigma )+|(f_n-f_m)(a,\beta (a))|<ϵ.\hfill \end{array}$$

Therefore, the succession ${(f_n(x,y))}_n$ is a Cauchy sequence of numbers and thus convergent. Note additionally that this convergence is uniform; therefore, we define the function $f:\sigma \to \mathbb{R}$ by $$f(\cdot ):=\underset{n\to \infty }{\lim }{f}_n(\cdot ).$$

We need to verify if the function this way defined is in $BV(\sigma ).$

Let’s see. If $\eta ={\left\{y_j\right\}}_{j=0}^s$ is a partition of [e, d], we will have that, $$\begin{array}{cc}\sum _{j=1}^s\left|f\left(\alpha (y_j),y_j\right)-f\left(\alpha (y_{j-1}),y_{j-1}\right)\right|\hfill & \le \sum _{i=1}^s\left|f\left(\alpha (y_j),y_j\right)-f_N\left(\alpha (y_j),y_j\right)\right|\hfill \\ +\sum _{i=1}^s\left|f_N\left(\alpha (y_j),y_j\right)-f_N\left(\alpha (y_{j-1}),y_{j-1}\right)\right|\hfill & \hfill \\ +\sum _{i=1}^s\left|f_N\left(\alpha (y_{j-1}),y_{j-1}\right)-f\left(\alpha (y_{j-1}),y_{j-1}\right)\right|\hfill & \hfill \\ \le ϵ+V_{10}(f_N,\sigma ).\hfill & \hfill \end{array}$$

From where $$V_{01}(f,\sigma )\le ϵ+V_{01}(f_N,\sigma )\text{so}{V}_{01}(f,\sigma )<\infty .$$

On the other hand, by virtue of inequality (2.3)(2.3) $$TV(f,\mathbf{R})\le \underset{j\to \infty }{\lim \inf }TV(f_j,\mathbf{R})$$(2.3) , we will have that $$TV(f_v,\mathbf{R})\le \underset{j\to \infty }{\lim \inf }TV({(f_j)}_v,\mathbf{R}).$$

Therefore, $$\begin{array}{cc}{V}_{10}(f,\sigma )+V_{01}(f,\sigma )+V_{11}(f,\sigma )\hfill & =V_{10}(f_v,\mathbf{R})+V_{01}(f,\sigma )+V_{11}(f_v,\mathbf{R})\hfill \\ \le \underset{j\to \infty }{\lim \inf }TV({(f_j)}_v,\mathbf{R})+V_{01}(f,\sigma )<\infty .\hfill & \hfill \end{array}$$

Now, it is valid to ask the question, ”Why choose this extension? Could it be extended $f_h$ using $\alpha$ and A defined in (3.2)?” The answer is affirmative. It is easy to show that if $f_v$ has bounded variation on $\mathbf{R},$ then $f_h$ will also go have bounded variation. It is important to clarify that, in general, this two variations do not coincide.

5. Jordan-type decomposition theorem

It is known that any function of one variable that has bounded variation can be written as the difference of two non-decreasing functions. The idea of the following theorem is to emulate this result. Before we start, let’s present some definitions that will be necessary.

Definition 5.1

(see (Idczak, 1994)) A function $f:\mathbf{R}\to \mathbb{R}$ will be called non-decreasing if $f(\cdot ,c)$ and $f(a,\cdot )$ are non-decreasing and the associated function $F_f$ is nonnegative.

Theorem 5.2

If a function $f:\sigma \to \mathbb{R}$ has a finite variation then there exist non-decreasing functions $g,h:\sigma \to \mathbb{R}$ such that $f=g-h.$

Proof.

First, let us assume that $f:\sigma \to \mathbb{R}$ has a finite variation. Then, for each $(x,y)\in \sigma ,$ we have that $$\begin{array}{cc}f(x,y)=f_v(x,y)\hfill & \hfill \\ =f_v(a,c)-f_v(x,c)+f_v(x,y)-f_v(a,y)+(f_v(x,c)+f_v(a,y)-f_v(a,c))\hfill & \hfill \\ =F_f([a,x]\times [c,y])+(f_v(x,c)+f_v(a,y)-f_v(a,c)).\hfill & \hfill \end{array}$$

Now, there exist non-decreasing functions $g^1,h^1,g^2$ and $h^2$ of one variable such that $$\begin{array}{cc}{f}_v(x,a)\hfill & =g^1(x)-h^1(x),x\in [a,b]\hfill \\ f_v(c,y)\hfill & =g^2(y)-h^2(y),y\in [c,d].\hfill \end{array}$$

In addition, we can choose two rectangle functions F and G that are additive and non-negative such that $$F_{f_v}(I_{ij})=G(I_{ij})-H(I_{ij})\text{for all}{I}_{ij}\subseteq [a,b]\times [c,d].$$

Using this knowledge about functions defined on rectangles, we can now proceed: $$\begin{array}{cc}g(x,y)\hfill & =g^1(x)+g^2(y)-\frac{1}{2}{f}_v(a,c)+G\left([a,x]\times [c,y]\right),\hfill \\ h(x,y)\hfill & =h^1(x)+h^2(y)-\frac{1}{2}{f}_v(a,c)+H\left([a,x]\times [c,y]\right).\hfill \end{array}$$

Thus, $$f_v(x,y)=g(x,y)-h(x,y)\text{for}(x,y)\in [a,b]\times [c,d].$$

In particular, $$f(x,y)=g(x,y)-h(x,y)\text{for}(x,y)\in \sigma .$$

6. Conclusions

In this paper, we have presented a definition of variation for functions defined on sets that are more general than those defined in the plane, specifically on convex and compact sets. To do this, we extend the function defined to the smallest rectangle that contains the said set, thus leveraging the known properties of function variation over rectangles. Subsequently, properties of this definition are presented, revealing that a set composed of functions defined on a convex and compact set, whose variation is bounded, is sufficiently rich as it contains all Lipschitz functions. Consequently, we obtain a space of functions equipped with a norm, with which it is demonstrated to be a Banach space. Finally, a Jordan-type decomposition theorem is obtained, meaning that a function in this space can be expressed as the difference of functions.

Disclosure statement

The authors declare that they have no conflict of interest related to this research, either by authorship or publication. They also declare that they have not received specific funding from any source, whether commercial, public, or non-profit organizations.