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| In [[mathematics]], '''Fredholm theory''' is a theory of [[integral equation]]s. In the narrowest sense, Fredholm theory concerns itself with the solution of the [[Fredholm integral equation]]. In a broader sense, the abstract structure of Fredholm's theory is given in terms of the [[spectral theory]] of [[Fredholm operator]]s and [[Fredholm kernel]]s on [[Hilbert space]]. The theory is named in honour of [[Erik Ivar Fredholm]].
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| ==Overview==
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| The following sections provide a casual sketch of the place of Fredholm theory in the broader context of [[operator theory]] and [[functional analysis]]. The outline presented here is broad, whereas the difficulty of formalizing this sketch is, of course, in the details.
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| ==Homogeneous equations==
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| Much of Fredholm theory concerns itself with finding solutions for the [[integral equation]]
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| :<math>g(x)=\int_a^b K(x,y) f(y)\,dy.</math>
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| This equation arises naturally in many problems in [[physics]] and mathematics, as the inverse of a [[differential equation]]. That is, one is asked to solve the differential equation
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| :<math>Lg(x)=f(x)</math>
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| where the function ''f'' is given and ''g'' is unknown. Here, ''L'' stands for a linear [[differential operator]]. For example, one might take ''L'' to be an [[elliptic operator]], such as
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| :<math>L=\frac{d^2}{dx^2}\,</math>
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| in which case the equation to be solved becomes the [[Poisson equation]]. A general method of solving such equations is by means of [[Green's function]]s, namely, rather than a direct attack, one instead attempts to solve the equation
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| :<math>LK(x,y) = \delta(x-y)</math>
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| where <math>\delta(x)</math> is the [[Dirac delta function]]. The desired solution to the differential equation is then written as
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| :<math>g(x)=\int K(x,y) f(y)\,dy.</math>
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| This integral is written in the form of a [[Fredholm integral equation]]. The function <math>K(x,y)</math> is variously known as a Green's function, or the [[integral transform|kernel of an integral]]. It is sometimes called the '''nucleus''' of the integral, whence the term [[nuclear operator]] arises.
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| In the general theory, ''x'' and ''y'' may be points on any [[manifold]]; the [[real number line]] or ''m''-dimensional [[Euclidean space]] 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 function]]s is studied, and [[Sobolev space]]s appear often.
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| The actual function space used is often determined by the solutions of the [[eigenvalue]] problem of the differential operator; that is, by the solutions to
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| :<math>L\psi_n(x)=\omega_n \psi_n(x)</math>
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| where the <math>\omega_n</math> are the eigenvalues, and the <math>\psi_n(x)</math> are the eigenvectors. The set of eigenvectors span a [[Banach space]], and, when there is a natural [[inner product]], then the eigenvectors span a [[Hilbert space]], at which point the [[Riesz representation theorem]] is applied. Examples of such spaces are the [[orthogonal polynomials]] that occur as the solutions to a class of second-order [[ordinary differential equation]]s.
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| Given a Hilbert space as above, the kernel may be written in the form
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| :<math>K(x,y)=\sum_n \frac{\psi_n^*(x) \psi_n(y)} {\omega_n}</math>
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| where <math>\psi_n^*</math> is the [[dual space|dual]] to <math>\psi_n</math>. In this form, the object <math>K(x,y)</math> is often called the [[Fredholm operator]] or the [[Fredholm kernel]]. That this is the same kernel as before follows from the [[complete space|completeness]] of the basis of the Hilbert space, namely, that one has
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| :<math>\delta(x-y)=\sum_n \psi_n^*(x) \psi_n(y).</math>
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| Since the <math>\omega_n</math> are generally increasing, the resulting eigenvalues of the operator <math>K(x,y)</math> are thus seen to be decreasing towards zero.
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| ==Inhomogeneous equations==
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| The inhomogenous Fredholm integral equation
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| :<math>f(x)=- \omega \phi(x) + \int K(x,y) \phi(y)\,dy</math>
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| may be written formally as
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| :<math>f = (K-\omega) \phi</math>
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| which has the formal solution
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| :<math>\phi=\frac{1}{K-\omega} f.</math>
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| A solution of this form is referred to as the [[resolvent formalism]], where the resolvent is defined as the operator
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| :<math>R(\omega)= \frac{1}{K-\omega I}.</math>
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| Given the collection of eigenvectors and eigenvalues of ''K'', the resolvent may be given a concrete form as
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| :<math>R(\omega; x,y) = \sum_n \frac{\psi_n^*(y)\psi_n(x)}{\omega_n - \omega}</math>
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| with the solution being
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| :<math>\phi(x)=\int R(\omega; x,y) f(y)\,dy.</math>
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| A necessary and sufficient condition for such a solution to exist is one of [[Fredholm's theorem]]s. The resolvent is commonly expanded in powers of <math>\lambda=1/\omega</math>, in which case it is known as the [[Liouville-Neumann series]]. In this case, the integral equation is written as
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| :<math>g(x)= \phi(x) - \lambda \int K(x,y) \phi(y)\,dy</math>
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| and the resolvent is written in the alternate form as
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| :<math>R(\lambda)= \frac{1}{I-\lambda K}.</math>
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| ==Fredholm determinant==
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| The [[Fredholm determinant]] is commonly defined as
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| :<math>\det(I-\lambda K) = \exp \left[
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| -\sum_n \frac{\lambda^n}{n} \operatorname{Tr}\, K^n \right]</math>
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| where | |
| :<math>\operatorname{Tr}\, K = \int K(x,x)\,dx</math>
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| :<math>\operatorname{Tr}\, K^2 = \iint K(x,y) K(y,x) \,dx\,dy</math>
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| and so on. The corresponding [[Riemann zeta function|zeta function]] is
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| :<math>\zeta(s) = \frac{1}{\det(I-s K)}.</math>
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| The zeta function can be thought of as the determinant of the [[Resolvent formalism|resolvent]].
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| The zeta function plays an important role in studying [[dynamical systems]]. Note that this is the same general type of zeta function as the [[Riemann zeta function]]; however, in this case, the corresponding kernel is not known. The existence of such a kernel is known as the [[Hilbert–Pólya conjecture]].
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| ==Main results==
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| The classical results of the theory are [[Fredholm's theorem]]s, one of which is the [[Fredholm alternative]].
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| One of the important results from the general theory are that the kernel is a [[compact operator]] when the space of functions are [[equicontinuous]].
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| A related celebrated result is the [[Atiyah–Singer index theorem]], pertaining to index (dim ker – dim coker) of elliptic operators on [[compact manifold]]s.
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| ==History==
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| Fredholm's 1903 paper in ''Acta Mathematica'' is considered to be one of the major landmarks in the establishment of [[operator theory]]. [[David Hilbert]] developed the abstraction of [[Hilbert space]] in association with research on integral equations prompted by Fredholm's (amongst other things).
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| ==References==
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| * E.I. Fredholm, "Sur une classe d'equations fonctionnelles", ''Acta Mathematica'', '''27''' (1903) pp. 365–390.
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| * D.E. Edmunds and W.D. Evans (1987), ''Spectral theory and differential operators,'' Oxford University Press. ISBN 0-19-853542-2.
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| * {{springer|id=f/f041440|title=Fredholm kernel|author=B.V. Khvedelidze, G.L. Litvinov}}
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| * Bruce K. Driver, "[http://math.ucsd.edu/~driver/231-02-03/Lecture_Notes/compact.pdf Compact and Fredholm Operators and the Spectral Theorem]", ''Analysis Tools with Applications'', Chapter 35, pp. 579–600.
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| * Robert C. McOwen, "[http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1102780323 Fredholm theory of partial differential equations on complete Riemannian manifolds]", ''Pacific J. Math.'' '''87''', no. 1 (1980), 169–185.
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| ==See also==
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| *[[Green's functions]]
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| *[[Spectral theory]]
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| [[Category:Functional analysis]]
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