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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

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.

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 Let $f: \mathbb R\to\mathbb R$ be a $2\pi$ periodic square 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)