Green's function

Spekkie Big

Hi,

I'm trying to understand an article about thin plate splines. My understanding so far:

Suppose you have a infinite thin metal plate and set of M points {(x_1,y_1),...,(x_M, y_M)}
At each point in the set you want to constrain the metal plate to a certain height above or below the plane. So suppose f(x,y) represents the metal plate, then f(x_1,y_1) = z_1, ... ,
f(x_M, y_M) = z_M. Now you want to know f(x,y) for all other points (x,y) in the domain.

f(x,y) should be a solution to the biharmonic equation D^2 f = 0, because this equation governs the shape of a thin plate under an out-of-plane load. (D^2 is the biharmonic operator). Now in the article it is stated that U(r) = r^2 log r^2 is the fundamental solution (or Green's function) of D^2. ( r = sqrt( x^2 + y^2 ) )

It turns out that f(x,y) can be expressed as a linear combination of M copies of U(r) centered on each of the M points from the set.

so: f(x,y) = w1 * U( sqrt( (x - x_1)^2 + (y - y_1)^2 ) ) + ... + wM * U( sqrt( (x - x_M)^2 + (y - y_M)^2 ) )

the weights w1 .. wM can be determined by solving a system of equations by constraining f(x_1,y_1) to z_1 ... f(x_M, y_M) to z_M.

I understand that a linear combination of U(r)'s is still a solution to D^2 f = 0. What is don't get is how do you proceed from the Green's function to a solution of the biharmonic equation with constriants as described above? It's hard to find accessible explanations on the internet, so I hope someone could point me in the right direction.

Thanx.