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 ...
, a function of bounded variation, also known as ' function, is a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-valued
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
whose
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 ...
is bounded (finite): the
graph of a function having this property is well behaved in a precise sense. For a
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
of a single
variable
Variable may refer to:
* Variable (computer science), a symbolic name associated with a value and whose associated value may be changed
* Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
, being of bounded variation means that the
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
along the
direction of the
-axis, neglecting the contribution of motion along
-axis, traveled by a
point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Point ...
moving along the graph has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is a
hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
in this case), but can be every
intersection of the graph itself with a
hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...
(in the case of functions of two variables, a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
) parallel to a fixed -axis and to the -axis.
Functions of bounded variation are precisely those with respect to which one may find
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 ...
s of all continuous functions.
Another characterization states that the functions of bounded variation on a compact interval are exactly those which can be written as a difference , where both and are bounded
monotone
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony.
Monotone or monotonicity may also refer to:
In economics
*Monotone preferences, a property of a consumer's preference ordering.
*Monotonic ...
. In particular, a BV function may have discontinuities, but at most countably many.
In the case of several variables, a function defined on an
open subset of
is said to have bounded variation if its
distributional derivative
Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose de ...
is a
vector-valued finite
Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel ...
.
One of the most important aspects of functions of bounded variation is that they form an
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
of
discontinuous functions whose first derivative exists
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
: due to this fact, they can and frequently are used to define
generalized solution
In mathematics, a weak solution (also called a generalized solution) to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precise ...
s of nonlinear problems involving
functionals,
ordinary and
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s in
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
,
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
.
We have the following chains of inclusions for continuous functions over a closed, bounded interval of the real line:
:
Continuously differentiable ⊆
Lipschitz continuous
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there e ...
⊆
absolutely continuous
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central ope ...
⊆ continuous and bounded variation ⊆
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
History
According to Boris Golubov, ''BV'' functions of a single variable were first introduced by
Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...
, in the paper dealing with the convergence of
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
. The first successful step in the generalization of this concept to functions of several variables was due to
Leonida Tonelli
Leonida Tonelli (19 April 1885 – 12 March 1946) was an Italian mathematician, noted for creating Tonelli's theorem, a variation of Fubini's theorem, and for introducing semicontinuity methods as a common tool for the direct method in the calc ...
, who introduced a class of ''continuous'' ''BV'' functions in 1926 , to extend his
direct method for finding solutions to problems in the
calculus of variations in more than one variable. Ten years after, in ,
Lamberto Cesari
Lamberto Cesari (23 September 1910 – 12 March 1990) was an Italian mathematician naturalized in the United States, known for his work on the theory of surface area, the theory of functions of bounded variation, the theory of optimal control ...
''changed the continuity requirement'' in Tonelli's definition ''to a less restrictive
integrability requirement'', obtaining for the first time the class of functions of bounded variation of several variables in its full generality: as Jordan did before him, he applied the concept to resolve of a problem concerning the convergence of Fourier series, but for functions of ''two variables''. After him, several authors applied ''BV'' functions to study
Fourier series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
in several variables,
geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
, calculus of variations, and
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
.
Renato Caccioppoli
Renato Caccioppoli (; 20 January 1904 – 8 May 1959) was an Italian mathematician, known for his contributions to mathematical analysis, including the theory of functions of several complex variables, functional analysis, measure theory.
Life a ...
and
Ennio de Giorgi used them to define
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
of
nonsmooth boundaries of
sets (see the entry "''
Caccioppoli set''" for further information).
Olga Arsenievna Oleinik
Olga Arsenievna Oleinik (also as ''Oleĭnik'') HFRSE (russian: link=no, О́льга Арсе́ньевна Оле́йник) (2 July 1925 – 13 October 2001) was a Soviet mathematician who conducted pioneering work on the theory of partial di ...
introduced her view of generalized solutions for
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...
s as functions from the space ''BV'' in the paper , and was able to construct a generalized solution of bounded variation of a
first order partial differential equation in the paper : few years later,
Edward D. Conway and
Joel A. Smoller applied ''BV''-functions to the study of a single
nonlinear hyperbolic partial differential equation of first order in the paper , proving that the solution of the
Cauchy problem
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. A Cauchy problem can be an initial value problem or a boundary value prob ...
for such equations is a function of bounded variation, provided the
initial value
In multivariable calculus, an initial value problem (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or ot ...
belongs to the same class.
Aizik Isaakovich Vol'pert
Aizik Isaakovich Vol'pert (russian: Айзик Исаакович Вольперт) (5 June 1923 – January 2006) (the family name is also transliterated as Volpert or WolpertSee .) was a Soviet and Israeli mathematician and chemical engineer ...
developed extensively a calculus for ''BV'' functions: in the paper he proved the
chain rule for BV functions and in the book he, jointly with his pupil
Sergei Ivanovich Hudjaev, explored extensively the properties of ''BV'' functions and their application. His chain rule formula was later extended by
Luigi Ambrosio
Luigi Ambrosio (born 27 January 1963) is a professor at Scuola Normale Superiore in Pisa, Italy. His main fields of research are the calculus of variations and geometric measure theory.
Biography
Ambrosio entered the Scuola Normale Superiore ...
and
Gianni Dal Maso in the paper .
Formal definition
''BV'' functions of one variable
The
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 ...
of a continuous
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
-valued (or more generally
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
-valued)
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
''f'', defined on an
interval 'a'', ''b''nbsp;⊂ ℝ is the quantity
:
where the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
is taken over the set
of all
partitions
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
of the interval considered.
If ''f'' is
differentiable
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...
and its derivative is Riemann-integrable, its total variation is the vertical component of the
arc-length of its graph, that is to say,
:
A continuous real-valued function
on the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
is said to be of bounded variation (BV function) on a chosen
interval 'a'', ''b''nbsp;⊂ ℝ if its total variation is finite, ''i.e.''
:
It can be proved that a real function ''ƒ'' is of bounded variation in