The representer for the Dirac delta at the point y is K(x-y)=exp(-|x - y|)/(4*pi).
Note that K(x) is an RBF. Suppose that we are given a set of scattered
sites X = {x1, ..., xN} and that we know values
of a function f in W21(R) at these sites,
{f(x1), ..., f(xN)}. We can construct a unique
interpolant
IXf(x) = c1 K(x - x1) + ... +
cN K(x - xN)
for the data. The important thing here is that in the inner product
for W21(R), f - IXf is
orthogonal to
span{K(x - x1), ..., K(x - xN)}.
Since IXf is in this span, it follows that it is the
orthogonal projection of f onto this span, and that
|| f - IXf || = minb's||f - b1 K(· -
x1) + ... + bN K(· - xN||.
Note that this implies || f - IXf || <= || f ||. In
addition, we also have that
f(u) - IXf(u) = < f - IXf, K(· - u)
>
f(u) - IXf(u) = < f - IXf, K(· - u) -
b1 K(· - x1) - ... - bN
K(· - xN >
Hence, by Schwarz's inequality, we arrive at
|f(u) - IXf(u)| <= || f - IXf ||*|| K(· - u) -
b1 K(· - x1) - ... - bN
K(· - xN ||
<= || f || || K(· - u) -
b1 K(· - x1) - ... -
bNK(· - xN ||.
Now, we define the power function
PX(u) := minb's || K(· - u) -
b1 K(· - x1) - ... -
bNK(· - xN ||,
and we arrive at one of the standard error estimates in the RKHS
theory:
|f(u) - IXf(u)| <= || f || PX(u).
In class, we showed that PX(u) <=
(h/(2*pi))½, where h = distance(u,X).