In
mathematics, Fredholm theory is a theory of
integral equation
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...
s. In the narrowest sense, Fredholm theory concerns itself with the solution of the
Fredholm integral equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to so ...
. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
of
Fredholm operator
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : '' ...
s and
Fredholm kernel
In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral equation and the Fredholm operator, and are o ...
s on
Hilbert space. The theory is named in honour of
Erik Ivar Fredholm
Erik Ivar Fredholm (7 April 1866 – 17 August 1927) was a Swedish mathematician whose work on integral equations and operator theory foreshadowed the theory of Hilbert spaces.
Biography
Fredholm was born in Stockholm in 1866. He obtained his P ...
.
Overview
The following sections provide a casual sketch of the place of Fredholm theory in the broader context of
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators ...
and
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined o ...
. The outline presented here is broad, whereas the difficulty of formalizing this sketch is, of course, in the details.
Fredholm equation of the first kind
Much of Fredholm theory concerns itself with the following
integral equation
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ...
for ''f'' when ''g'' and ''K'' are given:
:
This equation arises naturally in many problems in
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 mathematics, as the inverse of a
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
. That is, one is asked to solve the differential equation
:
where the function is given and is unknown. Here, stands for a linear
differential operator.
For example, one might take to be an
elliptic operator, such as
:
in which case the equation to be solved becomes the
Poisson equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
.
A general method of solving such equations is by means of
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
s, namely, rather than a direct attack, one first finds the function
such that for a given pair ,
:
where is the
Dirac delta function.
The desired solution to the above differential equation is then written as an integral in the form of a
Fredholm integral equation In mathematics, the Fredholm integral equation is an integral equation whose solution gives rise to Fredholm theory, the study of Fredholm kernels and Fredholm operators. The integral equation was studied by Ivar Fredholm. A useful method to so ...
,
:
The function is variously known as a Green's function, or the
kernel of an integral. It is sometimes called the nucleus of the integral, whence the term
nuclear operator
In mathematics, nuclear operators are an important class of linear operators introduced by Alexander Grothendieck in his doctoral dissertation. Nuclear operators are intimately tied to the projective tensor product of two topological vector spac ...
arises.
In the general theory, and may be points on any
manifold; the
real number line
In elementary mathematics, a number line is a picture of a graduated straight 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 number to a poin ...
or -dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
in the simplest cases. The general theory also often requires that the functions belong to some given
function space: often, the space of
square-integrable functions is studied, and
Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense ...
s appear often.
The actual function space used is often determined by the solutions of the
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
problem of the differential operator; that is, by the solutions to
:
where the are the eigenvalues, and the are the eigenvectors. The set of eigenvectors span a
Banach space, and, when there is a natural
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, then the eigenvectors span a
Hilbert space, at which point the
Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, ...
is applied. Examples of such spaces are the
orthogonal polynomials
In mathematics, an orthogonal polynomial sequence is a family of polynomials such that any two different polynomials in the sequence are orthogonal to each other under some inner product.
The most widely used orthogonal polynomials are the class ...
that occur as the solutions to a class of second-order
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s.
Given a Hilbert space as above, the kernel may be written in the form
:
In this form, the object is often called the
Fredholm operator
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : '' ...
or the
Fredholm kernel
In mathematics, a Fredholm kernel is a certain type of a kernel on a Banach space, associated with nuclear operators on the Banach space. They are an abstraction of the idea of the Fredholm integral equation and the Fredholm operator, and are o ...
. That this is the same kernel as before follows from the
completeness of the basis of the Hilbert space, namely, that one has
:
Since the are generally increasing, the resulting eigenvalues of the operator are thus seen to be decreasing towards zero.
Inhomogeneous equations
The inhomogeneous Fredholm integral equation
:
may be written formally as
:
which has the formal solution
:
A solution of this form is referred to as the
resolvent formalism, where the resolvent is defined as the operator
:
Given the collection of eigenvectors and eigenvalues of ''K'', the resolvent may be given a concrete form as
:
with the solution being
:
A necessary and sufficient condition for such a solution to exist is one of
Fredholm's theorem In mathematics, Fredholm's theorems are a set of celebrated results of Ivar Fredholm in the Fredholm theory of integral equations. There are several closely related theorems, which may be stated in terms of integral equations, in terms of linear a ...
s. The resolvent is commonly expanded in powers of
, in which case it is known as the
Liouville-Neumann series. In this case, the integral equation is written as
:
and the resolvent is written in the alternate form as
:
Fredholm determinant
The
Fredholm determinant In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a tra ...
is commonly defined as
: