View source for Hilbert theory of integral equations

A general theory of linear integral equations of the second kind,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047420/h0474201.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

established by D. Hilbert [[#References|[1]]] on the basis of his theory of linear and bilinear forms in an infinite number of variables. The principal idea of the theory is as follows. Let there be given a complete orthonormal system of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047420/h0474202.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047420/h0474203.png" /> and let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047420/h0474204.png" /></td> </tr></table>

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Solving the integral equation (1) is then equivalent to solving the infinite system of linear algebraic equations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047420/h0474206.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>

Here only those solutions of the system for which

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are considered, i.e. the system is considered in a [[Hilbert space|Hilbert space]]. The study of (2) in a Hilbert space makes it possible to study the properties of equation (1).

Hilbert's theory of integral equations gives a foundation for the extremal properties of the eigen values of integral equations with a [[Hermitian kernel|Hermitian kernel]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D. Hilbert,   "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen" , Chelsea, reprint  (1953)</TD></TR></table>



====Comments====
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047420/h0474208.png" /> is Hermitian and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047420/h0474209.png" /> are chosen as a complete orthonormal system of eigen functions corresponding to the eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047420/h04742010.png" /> of the integral operator, then the system (2) becomes diagonal; solving it leads to the representation

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if the conditions of the [[Fredholm alternative|Fredholm alternative]] are fulfilled. If instead of (1) the more general equation

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is considered, then

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which holds also for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047420/h04742014.png" />, i.e. for the case of integral equations of the first kind (see [[Hilbert–Schmidt series|Hilbert–Schmidt series]]).

The important thing is to consider the concrete integral equation (1) as an abstract linear operator equation on the Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047420/h04742015.png" />, so that the whole theory of Hilbert spaces, of which the discussion in the main article above is just one aspect, is available.

For more information and additional references see [[Integral equation with symmetric kernel|Integral equation with symmetric kernel]].