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)