Main Index
Mathematical Analysis
Real Analysis
Subject Index
comment on the page
Let be a real valued function defined on an interval
. The least upper bound of the sums of the type
where runs over all partitions of
, is called the (total) variation
of
over
. For instance,
. This concept was introduced by C.Jordan [1] .
Function is called of bounded (or finite) variation if there is a number
such that
.
If is real valued then roughly saying
is of bounded variation means that the graph of
has finite arc length.
Theorem: If a real valued function defined on
a) has bounded derivation here, or
b) is monotone here, or
c) has a finite number of local extrema here and is continuous here,
then is of bounded variation.
The sum, the difference and the product of two functions of bounded variation is again a function of bounded variation. If the modulus of the denominator is larger than a positive constant then the quotient of two such function is also of bounded variation.
Theorem (Jordan decomposition of functions of bounded variation): A real valued function is of bounded variation if and only if
can be represented in the form
, where
and
are monotone increasing functions on
.
To prove this put , and
.
It follows that if is of bounded variation, then
is bounded and its discontinuities are all jump discontinuities (i.e. of the so called first kind
) and are countable in number.
All these basic properties of functions of bounded variation were already found by Jordan [1] , [2] .
Jordan was led to the concept of functions of bounded variation in the context of generalization of the Dirichlet criterion for the convergence of Fourier series of piecewise monotone functions:
Theorem (Dirichlet-Jordan): Let ->R be a bounded function.
1) If is of bounded variation on an interval
about
for some
, then the Fourier series of
evaluated at
converges to
.
2) If is continuous and of bounded variation, then the Fourier series of
converges uniformly to
.
In 1904 Lebesgue [3] proved the following result:
Theorem (Lebesgue decomposition of a function of bounded variation): If is a function of bounded variation on the interval
then it can be represented in the form
,
where is a absolutely continuous function,
is a singular function, and
is a jump function.
The Lebesgue decomposition is unique in some cases, e.g. if . ( [4] , [5] ).
If is absolutely continuous on
then its total variation is
A multidimensional extension requires an appropriate generalization of total variation for functions of several variables is needed. There more possibilities how to do this.
Let be a real valued function defined on
,
. Given a set of indices
, let
denote the value of
at the point in
whose
th coordinate is equal to
if
and is equal to
otherwise. For an
-dimensional subinterval
of
, let
be the alternating sum of the values of
at the vertices of
. The variation in the sense of Vitali of
on
is defined by
,
where the supremum is taken over all partitions of
into
-dimensional subintervals.
This notion was introduced by Vitali [6] , and later rediscovered by Lebesgue [7] and Fréchet [8] .
If all the involved partial derivatives are continuous on then
An analog of the above Jordan decomposition theorem says that a function is of bounded Vitali variation if and only if it can be represented in the form
where for both functions
and
all the sums
are non-negative.
For non-empty , let
denote the function on
obtained by fixing the
th argument of
to
whenever
, and letting the other arguments vary. Function
is said to be of bounded Hardy-Krause variation if
is of bounded Vitali variation, for all non-empty
.
This type of variation was introduced by Hardy [9] in the case in connection with his investigation of convergence of double Fourier expansion of functions of two variables.
For functions of a single variable, the notions of bounded Vitali variation and bounded Hardy-Krause variation coincide, and reduce to the usual definition of bounded variation. In multivariate case there are functions that are of bounded Vitali variation but not of bounded Hardy-Krause variation. Let be not of bounded variation. Define
by
. Then
for all rectangles
since
does not vary with
. Thus defined
is of bounded Vitali variation, but not of bounded Hardy-Krause variation.
There is a similar analog to the Jordan decomposition theorem also for functions of bounded Hardy-Krause variation as above.
[1] | Jordan, C. (1881). Sur la série de Fourier. C.R. Acad. Sci. Paris Sér. I Math., 92(5), 228-230. |
[2] | Jordan, C. (1893). Cours d'analyse , Vol. 1. Paris: Gauthier-Villars. |
[3] | Lebesgue, H. (1928). Leçons sur l'intégration et la récherche des fonctions primitives. Paris: Gauthier-Villars. |
[4] | Natanson, I. P. (1961). Theorie der Funktionen einer reellen Veränderlichen. (Translated from Russian). Frankfurt a. Main: H.Deutsch. |
[5] | Natanson, I. P. (1955, 1961). Theory of functions of a real variable. Vol. 1 & 2 (Translated from Russian). New York: Frederick Ungar Publishing Co.. |
[6] | Vitali, G. (1908). Sui gruppi di punti e sulle funzioni di variabili reali (Italian). Atti Accad. sci. Torino, 43, 229-246. |
[7] | Lebesgue, H. (1910). Sur l'intégration des fonctions discontinues. Ann. de l'Éc. Norm. (3), 27, 361-450. |
[8] | Fréchet, M. (1910). Extension au cas des intégrales multiples d'une définition de l'intégrale due à Stieltjes. Nouv. Ann. (4), 10, 241-256. |
[9] | Hardy, G. H. (1905). On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters.. Quart. J., 37, 53-79. |
Cite this web-page as:
Štefan Porubský: Variation of a Function.