Quadratic inequalities deduced from the theory of reproducing kernels (Q1089414)

Let \(H_ k\) be a functional Hilbert space over a set \(\Omega\) with a reproducing kernel k. Let F be a non-constant entire function with \(F^{(n)}(0)\geq 0\) \((n=0,1,2,...)\) and consider the positive-definite kernel \(K=F\circ k\) on \(\Omega\times \Omega\). According to the Aronszajn theory there is a unique functional Hilbert space \(H_ K\) over \(\Omega\) whose reproducing kernel is K. Moreover, there exists a relationship [the reviewer, Proc. Am. Math. Soc. 83, 279-285 (1981; Zbl 0466.30011)] between the spaces \(H_ k\) and \(H_ K\) which is expressed as \(\| F\circ f\|^ 2_{F\circ k}\leq F(\| f\|^ 2_ k)\) for every f in \(H_ k\). Here the author applies this relationship in the special case that k is induced from a positive-definite \(N\times N\) matrix A.