# Mercer theorem

From Encyclopedia of Mathematics

The bilinear series

of a Hermitian positive-definite continuous kernel on (cf. Integral equation with symmetric kernel; Kernel of an integral operator), where is the closure of a bounded domain in , converges absolutely and uniformly in to . Here the are the characteristic numbers of the kernel and the are the corresponding orthonormalized eigen functions. If a kernel satisfies the conditions of Mercer's theorem, then the integral operator ,

is nuclear (cf. Nuclear operator) and its trace can be calculated by the formula

Mercer's theorem can be generalized to the case of a bounded discontinuous kernel.

The theorem was proved by J. Mercer [1].

#### References

[1] | J. Mercer, Philos. Trans. Roy. Soc. London Ser. A , 209 (1909) pp. 415–446 |

[2] | J. Mercer, "Functions of positive and negative type, and their connection with the theory of integral equations" Proc. Roy. Soc. London Ser. A , 83 (1908) pp. 69–70 |

[3] | I.G. Petrovskii, "Lectures on the theory of integral equations" , Graylock (1957) (Translated from Russian) |

[4] | F.G. Tricomi, "Integral equations" , Interscience (1957) |

[5] | M.A. Krasnosel'skii, et al., "Integral operators in spaces of summable functions" , Noordhoff (1976) (Translated from Russian) |

#### Comments

#### References

[a1] | I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1977) |

[a2] | A.C. Zaanen, "Linear analysis" , North-Holland (1956) |

**How to Cite This Entry:**

Mercer theorem.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Mercer_theorem&oldid=11889

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