# Hilbert theory of integral equations

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

$$\tag{1 } \phi ( x) + \int\limits _ { a } ^ { b } K ( x, s) \phi ( s) ds = f ( x),$$

established by D. Hilbert  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 $\{ \omega _ {n} ( x) \}$ on the interval $( a, b)$ and let

$$\phi _ {p} = \int\limits _ { a } ^ { b } \phi ( t) \omega _ {p} ( t) dt,\ \ f _ {p} = \int\limits _ { a } ^ { b } f ( t) \omega _ {p} ( t) dt,$$

$$a _ {pq} = \int\limits _ { a } ^ { b } \int\limits _ { a } ^ { b } K ( x, t) \omega _ {p} ( t) \omega _ {q} ( t) dx dt.$$

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

$$\tag{2 } \phi _ {p} + \sum _ {q = 1 } ^ \infty a _ {pq} \phi _ {q} = f _ {p} ,\ \ p = 1, 2 ,\dots .$$

Here only those solutions of the system for which

$$\sum _ {p = 1 } ^ \infty \phi _ {p} ^ {2} < + \infty$$

are considered, i.e. the system is considered in a 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.

How to Cite This Entry:
Hilbert theory of integral equations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert_theory_of_integral_equations&oldid=47235
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article