Mercer's Theorem

The Mercer's theorem is a mathematical statement from the branch of functional analysis . It is named after the mathematician James Mercer and says that the integral kernel of a positive, self-adjoint integral operator can be represented as a convergent series over its eigenvalues ​​and eigenvectors.

statement

Let be a compact subset of . Furthermore, let a continuous function for which the condition for all true so that the by -defined integral operator ${\displaystyle \chi }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle k\in C(\chi \times \chi )}$ ${\displaystyle k(s,t)={\overline {k(t,s)}}}$ ${\displaystyle s,t\in \chi }$ ${\displaystyle T_{k}\colon L^{2}(\chi )\rightarrow L^{2}(\chi )}$

${\displaystyle (T_{k}(f))(\cdot )=\int _{\chi }k(\cdot ,t)f(t)dt}$

is self adjoint . Let also be the eigenvalues ​​of the integral operator counted according to their geometric multiplicity with associated eigenfunctions . If the operator is also positive, that is ${\displaystyle \lambda _{1},\lambda _{2},\ldots }$ ${\displaystyle T_{k}}$ ${\displaystyle \phi _{1},\phi _{2},\ldots }$ ${\displaystyle T_{k}}$

${\displaystyle {\begin{aligned}\forall f\in L^{2}(\chi ):\quad \int _{\chi \times \chi }k(s,t)f(s)f(t)dsdt\geq 0\,,\end{aligned}}}$

then applies

${\displaystyle {\begin{aligned}k(s,t)=\sum _{j=1}^{\infty }\lambda _{j}\phi _{j}(s)\phi _{j}(t)\,,\end{aligned}}}$

where the convergence is absolute and uniform .

literature

• Bernhard Schölkopf, Alex Smola: Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond (Adaptive Computation and Machine Learning) , MIT Press, Cambridge, MA, 2002, ISBN 0-262-19475-9 .
• Wladimir Wapnik: The Nature of Statistical Learning Theory , Springer Verlag, New York, NY, USA, 1995.