Value determining sets and bases of kernels for a functional Hilbert space

Abstract

If H_D is a Hilbert space of functions defined on a set D, H_D is a functional Hilbert space provided H_D possesses a reproducing kernel for every point in D; i. e., for every w in D, there exists a unique function K^w in H_D such that (f,K^w ) = f(w) for every f in H_D. A value determining set for H_D is a subset N of D such that the only function in H_D which takes the value zero at each point of N is the zero function, The first result in this paper is concerned with a relationship between value determining sets for a functional Hilbert space and the reproducing kernel functions in the space. If H_D is a functional Hilbert space, there exists a value determining set N for H_D such that the span of the set K_N, the set of kernels K^w determined by the points w in N, is dense in H_D; furthermore, if H_D is a separable (finite or infinite dimensional) functional Hilbert space, there exists a value determining set N for H_D such that the cardinality of N is equal to the dimension of the space, and such that the span of the set K_N is dense in H_D. By using the results stated above, it is possible to obtain results which answer certain interpolation problems. If H_D is a finite dimensional functional Hilbert space of dimension n, and N = {w(i)} for i=1 to n is a value determining set for H_D, and if {a(i)} for i =1 to n is a set of complex scalars, there exists a unique function f in H_D such that f(w(i)) = a(i) for every i = 1 , 2, . . . , n. Let H_D be a separable (infinite dimensional) functional Hilbert space. Let N be a countable value determining set for H_D such that the span of the set K_N is dense in H_D. Let {Phi(k)} for k=1 to infinity be the orthonormal basis for H_D obtained by normalizing the orthogonal sequence obtained from K_N by the Gram-Schmidt orthogonalization process. Let {a(k)} for k=1 to infinity be a sequence of complex scalars, and define the sequence {b(k)} for k = 1 to infinity as follows: b(1) = 1/Phi(1)(w(1)) a(1) b(k+1) = 1/Phi(k+1)(w(k+1)) {a(k+1) - Summation from j=1 to k of b(j)Phi(j)(w(k+1))} for k >= 1. If Summation from k=1 to infinity l b(k) I < infinity, there exists a unique function f in H_D such that f(w(k)) = a(k) for all k = 1, 2, . . . .