RADIAL POSITIVE DEFINITE FUNCTIONS AND SCHOENBERG MATRICES WITH NEGATIVE EIGENVALUES

The main object under consideration is a class Phi(n)\Phi(n+1) of radial positive definite functions on R-n which do not admit radial positive definite continuation on Rn +1. We find certain necessary and sufficient conditions on the Schoenberg representation measure nu(n) of integral is an element of Phi(n) for integral is an element of Phi(n+k), k is an element of N. We show that the class Phi(n)\ Phi(n+k) is rich enough by giving a number of examples. In particular, we give a direct proof of Omega(n) is an element of Phi(n)\Phi(n+1), which avoids Schoenberg's theorem; Omega(n) is the Schoenberg kernel. We show that Omega(n)(a.) Omega(n) (b.) is an element of Phi(n)\Phi(n+1) for a not equal b. Moreover, for the square of this function we prove the surprisingly much stronger result Omega(2)(n)(a.) is an element of Phi(2n-1)\Phi(2n). We also show that any f is an element of Omega(n)\Phi(n+1), n >= 2, has infinitely many negative squares. The latter means that for an arbitrary positive integer N there is a finite Schoenberg matrix S-X(f) := parallel to f(vertical bar x(i) - x(j)vertical bar(n+1))parallel to(m)(i, j= 1), X := {x(j)}(j = 1)(m) subset of Rn+1, which has at least N negative eigenvalues.