Difference between revisions of "Variation of a function"
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961104.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961104.png" /></td> </tr></table> | ||
− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961105.png" /> is an arbitrary system of points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961106.png" />. This definition was given by C. Jordan [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961107.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961108.png" /> has (is of) bounded (finite) variation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961109.png" />, and the class of all such functions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611010.png" />. A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611011.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611012.png" /> if and only if it can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611015.png" /> are functions which increase on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611016.png" /> (the [[Jordan decomposition|Jordan decomposition]] of functions of bounded variation). The sum, the difference and the product of two functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611017.png" /> are also functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611018.png" />. This is also true of the quotient of two functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611019.png" /> if the modulus of the denominator is larger than a positive constant on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611020.png" />. Every function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611021.png" /> is bounded and cannot have more than a countable set of discontinuity points, all of which are of the first kind. All these properties of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611022.png" /> were established by Jordan [[#References|[1]]] (see also [[#References|[2]]]). | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961105.png" /> is an arbitrary system of points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961106.png" /> (cp. with [[Function of bounded variation#Total variation|Function of bounded variation]]. This definition was given by C. Jordan [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961107.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961108.png" /> has (is of) bounded (finite) variation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961109.png" />, and the class of all such functions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611010.png" />. A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611011.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611012.png" /> if and only if it can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611015.png" /> are functions which increase on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611016.png" /> (the [[Jordan decomposition|Jordan decomposition]] of functions of bounded variation). The sum, the difference and the product of two functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611017.png" /> are also functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611018.png" />. This is also true of the quotient of two functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611019.png" /> if the modulus of the denominator is larger than a positive constant on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611020.png" />. Every function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611021.png" /> is bounded and cannot have more than a countable set of discontinuity points, all of which are of the first kind. All these properties of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611022.png" /> were established by Jordan [[#References|[1]]] (see also [[#References|[2]]]). |
Functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611024.png" /> are almost-everywhere differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611025.png" /> and may be represented as | Functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611024.png" /> are almost-everywhere differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611025.png" /> and may be represented as |
Revision as of 15:15, 9 September 2012
A numerical characteristic of functions of one real variable which is connected with differentiability properties.
1) Let be a complex-valued function defined on an interval
; its variation
is the least upper bound of sums of the type
![]() |
where is an arbitrary system of points on
(cp. with Function of bounded variation. This definition was given by C. Jordan [1]. If
, one says that
has (is of) bounded (finite) variation over
, and the class of all such functions is denoted by
. A real-valued function
belongs to the class
if and only if it can be represented in the form
, where
and
are functions which increase on
(the Jordan decomposition of functions of bounded variation). The sum, the difference and the product of two functions of class
are also functions of class
. This is also true of the quotient of two functions of class
if the modulus of the denominator is larger than a positive constant on
. Every function in
is bounded and cannot have more than a countable set of discontinuity points, all of which are of the first kind. All these properties of functions in
were established by Jordan [1] (see also [2]).
Functions in
are almost-everywhere differentiable on
and may be represented as
![]() |
where is an absolutely continuous function,
is a singular function and
is a saltus function (the Lebesgue decomposition of a function of bounded variation). Such a decomposition is unique if
[3], [2].
The class was originally introduced by Jordan in the context of the generalization of the Dirichlet criterion for the convergence of Fourier series of piecewise-monotone functions. It was shown by him that Fourier series of
-periodic functions in the class
converge at all points of the real axis. Functions of bounded variation subsequently found extensive application in various branches of mathematics, especially in the theory of the Stieltjes integral.
One sometimes also considers classes , defined as follows. Let
(
,
) be a continuous function which increases monotonically if
. Let
be the least upper bound of sums of the type
![]() |
where is an arbitrary system of points in
. The quantity
is called the
-variation of
on
. If
, one says that
has bounded
-variation on
, while the class of such functions is denoted by
([4]). If
, one obtains Jordan's class
, while if
,
, one obtains Wiener's classes
[5]. The definition of the class
was proposed by L.C. Young [6].
If
![]() |
then
![]() |
In particular, on any interval ,
![]() |
for ,
, these being proper inclusions.
References
[1] | C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris Sér. I Math. , 92 : 5 (1881) pp. 228–230 |
[2] | I.P. Natanson, "Theory of functions of a real variable" , 1–2 , F. Ungar (1955–1961) (Translated from Russian) |
[3] | H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928) |
[4] | N.K. [N.K. Bari] Bary, "A treatise on trigonometric series" , Pergamon (1964) (Translated from Russian) |
[5] | N. Wiener, "The quadratic variation of a function and its Fourier coefficients" J. Math. and Phys. , 3 (1924) pp. 72–94 |
[6] | L.C. Young, "Sur une généralisation de la notion de variation de puissance ![]() |
Comments
The variation of a function as defined above is often called the total variation. It is the sum of the negative and positive variations (cf. Negative variation of a function; Positive variation of a function). One has
![]() |
where is the Banach indicatrix of
. If
, then
![]() |
References
[a1] | E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) pp. 266; 270; 272 |
[a2] | S. Saks, "Theory of the integral" , Hafner (1937) (Translated from French) |
Several different definitions of variation exist for functions of several variables (Arzelà variation; Vitali variation; Pierpont variation; Tonelli plane variation; Fréchet variation; Hardy variation). The following definition, [1], based on the use of the Banach indicatrix, also proved very fruitful. Let a real-valued function be given and be Lebesgue-measurable on an
-dimensional cube
. The variation
of order
, where
, of
on
is the number
![]() |
where denotes the
-th variation of the set
(cf. Variation of a set), while the integral is understood in the sense of Lebesgue. This definition allows one to transfer many properties of functions of bounded variation in one variable to functions of several variables. For instance,
a) ;
![]() |
b) If a sequence of functions ,
converges uniformly to
in
, then
![]() |
c) If the function is continuous in
and all its variations are finite,
has a total differential almost-everywhere.
d) If the function is absolutely continuous in
, then
![]() |
e) If the function is continuous in a cube
with side-length
, if it has bounded variations of all orders in
and if it can be periodically extended with period
for all arguments
,
, in the
-dimensional space, then its Fourier series converges uniformly to it on
(Pringsheim's theorem).
A sufficient condition for being of bounded variation is: If the function has continuous derivatives of all orders up to and including
in the cube
, then its variation of order
is finite. This theorem is a final theorem in the sense that the smoothness conditions cannot be improved for any
.
References
[1] | A.G. Vitushkin, "On higher-dimensional variations" , Moscow (1955) (In Russian) |
A.G. Vitushkin
Variation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variation_of_a_function&oldid=27856