# Degenerate kernel

From Encyclopedia of Mathematics

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.

#### Comments

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.

#### References

[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) |

**How to Cite This Entry:**

Degenerate kernel.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Degenerate_kernel&oldid=33817

This article was adapted from an original article by A.B. Bakushinskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article