Cauchy's functional equation is the
functional equation
In mathematics, a functional equation
is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning ...
:
A function
that solves this equation is called an
additive function
In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer variable ''n'' such that whenever ''a'' and ''b'' are coprime, the function applied to the product ''ab'' is the sum of the values of the func ...
. Over the
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s, it can be shown using
elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entail ...
that there is a single family of solutions, namely
for any rational constant
Over the
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, the family of
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s
now with
an arbitrary real constant, is likewise a family of solutions; however there can exist other solutions not of this form that are extremely complicated. However, any of a number of regularity conditions, some of them quite weak, will preclude the existence of these
pathological
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in th ...
solutions. For example, an additive function
is
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
if:
*
is
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
(
proven
Proven is a rural village in the Belgian province of West Flanders, and a "deelgemeente" of the municipality Poperinge. The village has about 1400 inhabitants.
The church and parish of Proven are named after Saint Victor. The Saint Victor Churc ...
by
Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
in 1821). This condition was weakened in 1875 by
Darboux Darboux is a surname. Notable people with the surname include:
*Jean Gaston Darboux
Jean-Gaston Darboux FAS MIF FRS FRSE (14 August 1842 – 23 February 1917) was a French mathematician.
Life
According this birth certificate he was bor ...
who showed that it is only necessary for the
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 ...
to be continuous at one point.
*
is
monotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
on any
interval.
*
is
bounded on any interval.
*
is
Lebesgue measurable
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
.
On the other hand, if no further conditions are imposed on
then (assuming the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
) there are infinitely many other functions that satisfy the equation. This was proved in 1905 by
Georg Hamel
Georg Karl Wilhelm Hamel (12 September 1877 – 4 October 1954) was a German mathematician with interests in mechanics, the foundations of mathematics and function theory.
Biography
Hamel was born in Düren, Rhenish Prussia. He studied at A ...
using
Hamel bases
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
. Such functions are sometimes called ''Hamel functions''.
The
fifth problem on
Hilbert's list is a generalisation of this equation. Functions where there exists a real number
such that
are known as Cauchy-Hamel functions and are used in Dehn-Hadwiger invariants which are used in the extension of
Hilbert's third problem
The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely m ...
from 3D to higher dimensions.
This equation is sometimes referred to as Cauchy's additive functional equation to distinguish it from
Cauchy's exponential functional equation Cauchy's logarithmic functional equation and
Cauchy's multiplicative functional equation
Solutions over the rational numbers
A simple argument, involving only elementary algebra, demonstrates that the set of additive maps
, where
are vector spaces over an extension field of
, is identical to the set of
-linear maps from
to
.
Theorem: ''Let
be an additive function. Then
is
-linear.''
Proof: We want to prove that any solution
to Cauchy’s functional equation,
, satisfies
for any
and
. Let
.
First note
, hence
, and therewith
from which follows
.
Via induction,
is proved for any
.
For any negative integer
we know
, therefore
. Thus far we have proved
:
for any
.
Let
, then
and hence
.
Finally, any
has a representation
with
and
, so, putting things together,
:
, q.e.d.
Properties of nonlinear solutions over the real numbers
We prove below that any other solutions must be highly
pathological
Pathology is the study of the causes and effects of disease or injury. The word ''pathology'' also refers to the study of disease in general, incorporating a wide range of biology research fields and medical practices. However, when used in th ...
functions.
In particular, it is shown that any other solution must have the property that its
graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
is
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
in
that is, that any disk in the plane (however small) contains a point from the graph.
From this it is easy to prove the various conditions given in the introductory paragraph.
Existence of nonlinear solutions over the real numbers
The linearity proof given above also applies to
where
is a scaled copy of the rationals. This shows that only linear solutions are permitted when the
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
of
is restricted to such sets. Thus, in general, we have
for all
and
However, as we will demonstrate below, highly pathological solutions can be found for functions
based on these linear solutions, by viewing the reals as a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
over the
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
of rational numbers. Note, however, that this method is nonconstructive, relying as it does on the existence of a
(Hamel) basis for any vector space, a statement proved using
Zorn's lemma. (In fact, the existence of a basis for every vector space is logically equivalent to the
axiom of choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
.)
To show that solutions other than the ones defined by
exist, we first note that because every vector space has a basis, there is a basis for
over the field
i.e. a set
with the property that any
can be expressed uniquely as
where
is a finite
subset of
and each
is in
We note that because no explicit basis for
over
can be written down, the pathological solutions defined below likewise cannot be expressed explicitly.
As argued above, the restriction of
to
must be a linear map for each
Moreover, because
for
it is clear that
is the constant of proportionality. In other words,
is the map
Since any
can be expressed as a unique (finite) linear combination of the
s, and
is additive,
is well-defined for all
and is given by:
It is easy to check that
is a solution to Cauchy's functional equation given a definition of
on the basis elements,
Moreover, it is clear that every solution is of this form. In particular, the solutions of the functional equation are linear
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is b ...
is constant over all
Thus, in a sense, despite the inability to exhibit a nonlinear solution, "most" (in the sense of cardinality
[It can easily be shown that ; thus there are functions each of which could be extended to a unique solution of the functional equation. On the other hand, there are only solutions that are linear.]) solutions to the Cauchy functional equation are actually nonlinear and pathological.
See also
*
*
*
*
References
*
External links
* Solution to the Cauchy Equatio
Rutgers University* {{cite web, url=https://math.stackexchange.com/q/423492, title=Overview of basic facts about Cauchy functional equation, website=StackExchange, date=2013, accessdate=20 December 2015, author=Martin Sleziak, display-authors=etal
Functional equations