Hamzi - Eurandom

Reproducing Kernel Hilbert Spaces
• Mercer Theorem: Let (X , µ) be a finite-measure space, and suppose
k ∈ L∞(X 2 ) is a symmetric real-valued function such that the integral
operator
Tk : L2(X ) → L2(X )
∫
f ↦→ (Tkf)(x) =
k(x, x
X
′ )f(x ′ )dµ(x ′ )
is positive definite; that is, for all f ∈ L2(X ), we have
∫
X 2 k(x, x ′ )f(x)f(x ′ )dµ(x)dµ(x ′ ) ≥ 0.
Let Ψj ∈ L2(X ) be the normalized orthogonal eigenfunctions of Tk
associated with the eigenvalues λj > 0, sorted in non-increasing order.
Then
i. (λj)j ∈ ℓ1,
ii. k(x, x ′ ) = ∑ NX
j=1 λjΨj(x)Ψj(x ′ ) holds for almost all (x, x ′ ). Either
NX ∈ N, or NX = ∞; in the latter case, the series converges
absolutely and uniformly for almost all (x, x ′ ).
. . . . . .
Boumediene Hamzi (Imperial College) On Control and RDS in RKHS June 4th, 2012 22 / 55