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 [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 $ \{ \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.

References

[1]	D. Hilbert, "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen" , Chelsea, reprint (1953)

Comments

If $ K $ is Hermitian and the $ \{ \omega _ {n} \} $ are chosen as a complete orthonormal system of eigen functions corresponding to the eigen values $ \lambda _ {n} $ of the integral operator, then the system (2) becomes diagonal; solving it leads to the representation

$$ \phi = \sum _ { p= } 1 ^ \infty \frac{f _ {p} }{1+ \lambda _ {p} } \omega _ {p} $$

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

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

is considered, then

$$ \phi = \sum _ { p= } 1 ^ \infty \frac{f _ {p} }{1+ \lambda _ {p} } \omega _ {p} , $$

which holds also for $ \lambda = 0 $, i.e. for the case of integral equations of the first kind (see Hilbert–Schmidt series).

The important thing is to consider the concrete integral equation (1) as an abstract linear operator equation on the Hilbert space $ L _ {2} [ a, b ] $, 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.