Degenerate kernel

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A kernel of a linear Fredholm integral operator (cf. Fredholm operator) of the form $$ \sum_{i=1}^N \phi_i(P) \psi_i(Q) $$ where $P$ and $Q$ are points in Euclidean spaces.


A linear Fredholm integral operator with a square-integrable kernel, considered as an operator on $L_2$, has a finite-dimensional range if and only if its kernel is degenerate.

Linear Fredholm integral equations of the second kind can be reduced to a system of linear algebraic equations if the kernel is degenerate.


[a1] H. Hochstadt, "Integral equations" , Wiley (Interscience) (1973) pp. Sect. 2.5
[a2] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)
[a3] L.V. Kantorovich, V.I. Krylov, "Approximate methods of higher analysis" , Noordhoff (1958) (Translated from Russian)
[a4] P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1958) (Translated from Russian)
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Degenerate kernel. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article