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Hilbert-Schmidt series

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A series of functions

(*)

where is the sequence of all eigen values of a symmetric kernel (cf. Kernel of an integral operator) , , is the corresponding sequence of orthonormal eigen functions, while is the scalar product of an arbitrary square-summable function and the function .

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

If there exists a constant C such that for all from the inequality

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


Comments

References

[a1] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)
How to Cite This Entry:
Hilbert-Schmidt series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hilbert-Schmidt_series&oldid=22577
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article