Hilbert-Schmidt series

A series of functions

$$ \tag{* } \sum _ {n = 1 } ^ \infty \frac{( f, \phi _ {n} ) }{\lambda _ {n} } \phi _ {n} ( x), $$

where $ \{ \lambda _ {n} \} $ is the sequence of all eigen values of a symmetric kernel (cf. Kernel of an integral operator) $ K ( x, s) $, $ a \leq x, s \leq b $, $ \{ \phi _ {n} ( x) \} $ is the corresponding sequence of orthonormal eigen functions, while $ ( f, \phi _ {n} ) $ is the scalar product of an arbitrary square-summable function $ f $ and the function $ \phi _ {n} $.

The Hilbert–Schmidt theorem: If the kernel $ K( x, s) $ is a square-summable function in two variables, then the series (*) converges in the mean to the function

$$ \int\limits _ { a } ^ { b } K ( x, s) f ( s) ds. $$

If there exists a constant C such that for all $ x $ from $ ( a, b) $ the inequality

$$ \int\limits _ { a } ^ { b } | K ( x, s) | ^ {2} ds \leq C $$

is fulfilled, then the Hilbert–Schmidt series converges absolutely and uniformly.

References

[a1]	I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)