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2020 Mathematics Subject Classification: Primary: 26A45 [MSN][ZBL] (Functions of one variable)

2020 Mathematics Subject Classification: Primary: 26B30 Secondary: 28A1526B1549Q15 [MSN][ZBL] (Functions of severable variables)

Functions of one variable

Classical definition

Let $I\subset \mathbb R$ be an interval. A function $f: I\to \mathbb R$ is said to have bounded variation if its total variation is bounded. The total variation is defined in the following way.

Definition 1 Let $I\subset \mathbb R$ be an interval and consider the collection $\Pi$ of ordered $2N$-ples points $a_1<b_1<a_2< b_2 < \ldots < a_N<b_N\in I$, where $N$ is an arbitrary natural number. The total variation of a function $f: I\to \mathbb R$ is given by \begin{equation}\label{e:TV} TV\, (f) := \sup \left\{ \sum_{i=1}^N |f(b_i)-f(a_i)| : (a_1, \ldots, b_N)\in\Pi\right\}\, \end{equation} (cp. with Section 4.4 of [Co] or Section 10.2 of [Ro]).

Generalizations

The definition of total variation of a function of one real variable can be easily generalized when the target is a metric space $(X,d)$: it suffices to substitute $|f(b_i)-f(a_i)|$ with $d (f(a_i), f(b_i))$ in \ref{e:TV}. Consequently, one defines functions of bounded variation taking values in an arbitrary metric space. Observe that, if $f:I\to X$ is a function of bounded variation and $\varphi:X\to Y$ a Lipschitz map, then $\varphi\circ f$ is also a function of bounded variation and \[ TV\, (\varphi\circ f) \leq {\rm Lip (\varphi)}\, TV\, (f)\, , \] where ${\rm Lip}\, (\varphi)$ denotes the Lipschitz constant of $\varphi$.

As a corollary we derive

Proposition 2 A function $(f^1, \ldots, f^k) = f: I\to \mathbb R^k$ is of bounded variation if and only if each coordinate function $f^j$ is of bounded variation.

General properties

Jordan decomposition

A fundamental characterization of functions of bounded variation of one variable is due to Jordan.

Theorem 3 Let $I\subset \mathbb R$ be an interval. A function $f: I\to\mathbb R$ has bounded variation if and only if it can be written as the difference of two bounded nondecreasing functions.

(Cp. with Theorem 4 of Section 5.2 in [Ro]). Indeed it is possible to find a canonical representation of any function of bounded variation as difference of nondecreasing functions.

Theorem 4 If $f:[a,b] \to\mathbb R$ is a function of bounded variation then there is a pair of nondecreasing functions $f^+$ and $f^-$ such that $f= f^+- f^-$ and $TV (f) = f^+ (b)-f^+ (a) + f^- (b)- f^- (a)$. The pair is unique up to addition of a constant, i.e. if $g^+$ and $g^-$ is a second pair with the same property, then $g^+-g^-=f^+-f^-\equiv {\rm const}$.

(Cp. with Theorem 3 of Section 5.2 in [Ro]). The latter representation of a function of bounded variation is also called Jordan decomposition.

Continuity

It follows immediately from Theorem 3 that

Proposition 5 If $f:I\to [a,b]$ is a function of bounded variation, then

• The right and left limits

\[ f (x^+) :=\lim_{y\downarrow x} f (y) \qquad f (x^-):= \lim_{y\uparrow x} f(y) \] exist at every point $x\in I$;

• The set of points of discontinuity of $f$ is at most countable.

Warning 6 However, according to the definitions given above, it may happen that at a goven point right and left limits coincide, but nonetheless the function $f$ is discontinuous. For instance the function $f:\mathbb R\to\mathbb R$ given by \[ f (x) =\left\{\begin{array}{ll} 1 \qquad &\mbox{if '"`UNIQ-MathJax35-QINU`"'}\\ 0 \qquad &\mbox{otherwise} \end{array}\right. \] is a function of bounded variation

Precise representative

In order to avoid patologies as in Warning 6 it is customary to postulate some additional assumptions for functions of bounded variations. Two popular choices are

• the imposition of right (resp. left) continuity, i.e. at any point $x$ we impose $f(x)=f (x^+)$ (resp. $f(x)=f(x^-$), cp. with Section 4.4 of [Co];
• at any point $x$ we impose $f(x) =\frac{1}{2} (f(x^+) + f(x^-))$.

The latter is perhaps more popular because of the Jordan criterion (see Theorem 11 below) and it is often called precise representative.

Differentiability

Functions of bounded variation of one variable are classically differentiable at a.e. point of their domain of definition, cp. with Corollary 5 of Section 5.2 in [Ro]. It turns out that such derivative is always a summable function (see below in the section Structure theorem). However, the fundamental theorem of calculus does not apply in this case, i.e. there are functions $f:[a,b]\to\mathbb R$ of bounded variation such that the identity \[ f(b') - f(a') =\int_{a'}^{b'} f' (t)\, dt \] fails for a set of pairs $(b', a')\in I\times I$ of positive measure (see below in the section Examples).

Measure theoretic characterization

Classically right-continuous functions of bounded variations can be mapped one-to-one to signed measures. More precisely, consider a signed measure $\mu$ on (the Borel subsets of) $\mathbb R$ with finite total variation (see Signed measure for the definition). We then define the function \begin{equation}\label{e:F_mu} F_\mu (x) := \mu (]-\infty, x])\, . \end{equation}

Theorem 7

• For every signed measure $\mu$ with finite total variation, $F_\mu$ is a right-continuous function of bounded variation such that $\lim_{x\to -\infty} F_mu (x) = 0$ and $TV (f)$ equals the total variation of $|\mu|$.
• For every right-continuous function $f:\mathbb R\to \mathbb R$ of bounded variations such that $\lim_{x\to-\infty} f (x) = 0$ there is a unique signed measure $\mu$ such that $f=F_\mu$

For a proof see Section 4 of Chapter 4 in [Co]. Obvious generalizations hold in the case of different domains of definition.

Distributional derivatives: modern definition

The measure $\mu$ is indeed the generalized derivatie of the function $f=F_\mu$ in the sense of distributions. More precisely \begin{equation}\label{e:distrib} \int f(t)\varphi' (t)\, dt = -\int \varphi (t)\, d\mu (t) \qquad \forall \varphi\in C^\infty_c (\mathbb R)\, . \end{equation} This identty is the starting point for the modern definition of functions of bounded variation, cp. with {{Cite|AFP} or Chapter 5 of [EG].

Definition 8 Let $I\subset\mathbb R$ be a bounded open interval. A function $f\in L^1 (E)$ is said to be of bounded variation if \begin{equation}\label{e:variation_modern} \sup \left\{ \int \varphi' (t) f(t)\, dt \;:\; \varphi\in C^\infty_c (I), \|\varphi\|_{C^0} \leq 1\right\} <\infty\, . \end{equation}

The following theorem links the classical and the modern definitions. See section 3.2 of [AFP] for a proof.

Theorem 9 Let $f$ and $I$ be as in Definition 8. Then there is a function $\tilde{f}:I\to\mathbb R$ and a signed measure $\mu$ on $I$ such that

• $\mu$ is the derivative, in the sense of distributions, of $f$, i.e. \eqref{e:distrib} holds
• $F_\mu = \tilde{f} = f$ almost everywhere
• $\tilde{f}$ is a function of bounded variation in the sense of Definition 1
• $TV (\tilde{f})$ equals the total variation of the measure $\mu$ which in turn is equal to the supremum in \eqref{e:variation_modern}.

Similar definitions and properties can be given for more general domains. However some caution is needed for unbounded domains since then functions of bounded variation are, in general, only locally summable.

Structure theorem

It is possible to relate the pointwise properties of a function $f: I\to \mathbb R$ of bounded variation with the properties of its generalized derivative $\mu$. More pecisely, using the Radon-Nikodym theorem we write $\mu = g \lambda + \mu_s$, where $\mu_s$ is a singular measure with respect to the Lebesgue measure $\mu$. We further follow the discussion of Section 3.2 of [AFP] and decompose $\mu_s = \mu_c +\mu_j$, where $\mu_c$ is the non-atomic part of the measure $\mu_s$, i.e. \[ \mu_c (\{x\}) = 0\qquad \mbox{for every '"`UNIQ-MathJax77-QINU`"'}\, \] and $\mu_j$ is the purely atomic part of $\mu_s$, that is, there is a set $J$ at most countable and weights $c_x\in \mathbb R, x\in J$ such that \[ \mu_j (E) = \sum_{x\in J\cap E} c_x\, . \] If we denote by $\delta_x$ the Dirac mass at the point $x$, then $\mu_j = \sum_{x\in J} c_x \delta_x$. We then have the following theorem (cp. with Section 3.2 of [AFP]), which is often referred to as BV structure theorem fur functions of one variable.

Theorem 10 Let $I = ]a,b[$, $f:I\to \mathbb R$ a right-continuous function of bounded variation and $\mu = g\lambda + \mu_c + \mu_j$.

• If $J$ denotes the set of points of discontinuity of $f$, then

\[ \mu_j = \sum_{x\in J} (f(x^+) - f(x^-)) \delta_x\, . \]

• At $\lambda$-a.e. $x$ the function $f$ is differentbaile and $f(x) = g(x)$.

Lebesgue decomposition

Observe also that, if we define the functions

• $f_a (x) := f(a)+ \int_a^x g(t)\, dt$,
• $f_j (x) := \mu_j (]a, x])$,
• $f_c (x) := \mu_c (]a, x])$,

then

• $f_a$ is an absolutely continuous function
• $f_c$ is a singular function
• $f_j$ is a jump function.

Then $f=f_a+f_c+f_j$ is called the Lebesgue decomposition of the function $f$ and it is unique up to constants.

Historical remark

Functions of bounded variation were introduced for the first time by C. Jordan in [Jo] to study the pointwise convergence of Fourier series. In particular Jordan proved the following generalization of Dirichlet theorem on the convergence of Fourier series, called Jordan criterion

Theorem 11 Let $f: \mathbb R\to\mathbb R$ be a $2\pi$ periodic summable function.

• If $f$ has bounded variation in an open interval $I$ then its Fourier series converges to $\frac{1}{2} (f (x^+) + f(x^-))$ at every $x\in I$.
• If in addition $f$ is continuous in $I$ then its Fourier series converges uniformly to $f$ on every closed interval $J\subset I$.

For a proof see Section 10.1 and Exercises 10.13 and 10.14 of [Ed]. The criterion is also called Jordan-Dirichlet test, see [Zy]

References

[AFP]	L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Co]	D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
[Ed]	R. E. Edwards, "Fourier series". Vol. 1. Holt, Rineheart and Winston, 1967.
[EG]	L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Ha]	P.R. Halmos, "Measure theory" , v. Nostrand (1950) MR0033869 Zbl 0040.16802
[HS]	E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) MR0188387 Zbl 0137.03202
[Jo]	C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230
[Ro]	H.L. Royden, "Real analysis" , Macmillan (1969) MR0151555 Zbl 0197.03501
[Zy]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988)