In
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, ''p''-variation is a collection of
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
s on functions from an ordered set to a
metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
, indexed by a real number
. ''p''-variation is a measure of the regularity or smoothness of a function. Specifically, if
, where
is a metric space and ''I'' a totally ordered set, its ''p''-variation is
:
where ''D'' ranges over all finite
partitions of the interval ''I''.
The ''p'' variation of a function decreases with ''p''. If ''f'' has finite ''p''-variation and ''g'' is an ''α''-Hölder continuous function, then
has finite
-variation.
The case when ''p'' is one is called
total variation
In mathematics, the total variation identifies several slightly different concepts, related to the ( local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval ...
, and functions with a finite 1-variation are called
bounded variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real-valued function whose total variation is bounded (finite): the graph of a function having this property is well behaved in a precise sense. For a conti ...
functions.
Link with Hölder norm
One can interpret the ''p''-variation as a parameter-independent version of the Hölder norm, which also extends to discontinuous functions.
If ''f'' is ''α''–
Hölder continuous Hölder:
* ''Hölder, Hoelder'' as surname
* Hölder condition
* Hölder's inequality
* Hölder mean
In mathematics, generalized means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of number ...
(i.e. its α–Hölder norm is finite) then its
-variation is finite. Specifically, on an interval
'a'',''b'' .
Conversely, if ''f'' is continuous and has finite ''p-''variation, there exists a reparameterisation,
, such that
is
Hölder continuous.
If ''p'' is less than ''q'' then the space of functions of finite ''p''-variation on a compact set is continuously embedded with norm 1 into those of finite ''q''-variation. I.e.
. However unlike the analogous situation with Hölder spaces the embedding is not compact.
For example, consider the real functions on
,1given by
. They are uniformly bounded in 1-variation and converge pointwise to a discontinuous function ''f'' but this not only is not a convergence in ''p''-variation for any ''p'' but also is not uniform convergence.
Application to Riemann–Stieltjes integration
If ''f'' and ''g'' are functions from
'a'', ''b''to ℝ with no common discontinuities and with ''f'' having finite ''p''-variation and ''g'' having finite ''q''-variation, with
then the
Riemann–Stieltjes Integral
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...
: