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INTRODUCTION

To explain Mercer's theorem, we first consider an important special case; see Below for a more general formulation.
A ''kernel'', in this context, is a continuous function that maps

: K: [a,b] imes [a,b] ightarrow \mathbb{R}

such that ''K''(''x'', ''s'') = ''K''(''s'', ''x'').

''K'' is said to be ''non-negative definite'' if and only if

: \sum_{i=1}^n\sum_{j=1}^n K(x_i, x_j) c_i c_j \geq 0

for all finite sequences of points ''x''1,...,''x''n of ''b'' and all choices of real numbers ''c''1, ..., ''c''n (cf. Positive Definite Kernel ).

Associated to ''K'' is a linear operator on functions defined by the integral

: arphi (x) =\int_a^b K(x,s) arphi(s)\, ds.

For technical considerations we assume φ can range through the space
''L''2 {Link without Title} (see Lp Space ) of square-integrable real-valued functions.
Since ''T'' is a linear operator, we can talk about Eigenvalues and Eigenfunction s of ''T''.

Theorem. Suppose ''K'' is a continuous symmetric non-negative definite kernel. Then there is an Orthonormal Basis
{''e''i}i of ''L''2 {Link without Title} consisting of eigenfunctions of ''T''''K'' such that the corresponding
sequence of eigenvalues {λi}i is nonnegative. The eigenfunctions corresponding to non-zero eigenvalues are continuous on ''b'' and ''K'' has the representation

: K(s,t) = \sum_{j=1}^\infty \lambda_j \, e_j(s) \, e_j(t)

where the convergence is absolute and uniform.


DETAILS


We now explain in greater detail the structure of the proof of
Mercer's theorem, particularly how it relates to Spectral Theory Of Compact Operators .

  • The map ''K'' → ''T''''K'' is injective.


  • ''T''''K'' is a non-negative symmetric compact operator on ''L''2 {Link without Title} ; moreover ''K''(''x'', ''x'') ≥ 0.


To show compactness, show that the image of the Unit Ball of ''L''2 under ''T''''K'' Equicontinuous and apply Ascoli's theorem, to show that the image of the unit ball is relatively compact in C([''a'',''b'' ) with the Uniform Norm and ''a fortiori'' in ''L''2[''a'',''b''].

Now apply the Spectral Theorem for compact operators on Hilbert
spaces to ''T''''K'' to show the existence of the
orthonormal basis {''e''i}i of
''L''2 {Link without Title}

: \lambda_i e_i(t)= e_i (t) = \int_a^b K(t,s) e_i(s)\, ds.

If λi ≠ 0, the eigenvector ''e''i is seen to be continuous on {Link without Title} . Now



where the convergence in the ''L''2 norm. Note that when continuity of the kernel is not assumed, the expansion no longer converges uniformly.


REFERENCES