ABSTRACT
For each point xj of the fundamental system {#i, ... , £<*n} we define the kernel polynomials gj E with axis xj, by
gj{x) = Gn(x,xj) = Gn(x ■ xj), j = l , . . . , d n. (5.3)
Define the vector valued function g : § r_1 —> Mdn by g(z) = [(¡fi(x) • • ■9dn{x)]T■ The corresponding basis matrix is
G - [g(»i) • • -g(®dn)]j Gij = Gn(xi • xj) for i , j = 1, . . . ,dn. (5.4)
Then the weights w such that (Anf)(x) = w • g(#) are given by the linear
system Gw = f, where f = [f(xi) • •' f{xdn)]T-Using this reproducing kernel basis the norm of the interpolation operator is
||A„|| = max ||G-1g(a:)||i. (5.5)
The value of Gn(l) is [15]
Gn(x,x) = G„(l) = V a: £ S2, (5.6)
so the matrix G in (5.4) has diagonal elements
r _ ( n + l )2 , \jrii — I — 1,. . . , an.
