Mercer theorem

The bilinear series

$$ \sum _ { m } \frac{\phi _ {m} ( s) \overline{ {\phi _ {m} ( t) }}\; }{ \lambda _ {m} } $$

of a Hermitian positive-definite continuous kernel $ K( s, t) $ on $ D \times D $( cf. Integral equation with symmetric kernel; Kernel of an integral operator), where $ D $ is the closure of a bounded domain in $ \mathbf R ^ {n} $, converges absolutely and uniformly in $ D \times D $ to $ K( s, t) $. Here the $ \lambda _ {m} $ are the characteristic numbers of the kernel $ K( s, t) $ and the $ \phi _ {m} ( s) $ are the corresponding orthonormalized eigen functions. If a kernel $ K $ satisfies the conditions of Mercer's theorem, then the integral operator $ T: L _ {2} ( D) \rightarrow L _ {2} ( D) $,

$$ Tf( s) = \int\limits _ { D } K( s, t) f( t) dt = \ \sum _ { m } \frac{1}{\lambda _ {m} } ( f, \phi _ {m} ) \phi _ {m} $$

is nuclear (cf. Nuclear operator) and its trace $ \sum _ {m} 1/ \lambda _ {m} $ can be calculated by the formula

$$ \sum _ { m } \frac{1}{\lambda _ {m} } = \ \int\limits _ { D } K( s, s) ds. $$

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)

References

[a1]	I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1977)
[a2]	A.C. Zaanen, "Linear analysis" , North-Holland (1956)