On positive-definite integral kernels and a related quadratic form.

Abstract

and B the dual space of C. B can be interpreted as the space of all complex Radon measures on [a, b], where the measure A acts on the function f to give ff(t)d,u(t) (all integrals are taken on [a, b] unless otherwise stated), and the dual norm on B can be interpreted as |||,u|| =total variation of /t. A continuous operator R from B to C is defined by R,(t) =fp(t, s)du(s). We shall assume that p is positive-definite, in the sense that ffp(s, t)dM(s)dp(t) > 0 for all nonzero ,u in B. Let H be L2[a, bl and let S be the restriction of R to those measures of the form dm=h(s)ds, hCH. The function p may also be interpreted as the covariance function of a Gaussian process defined with respect to: the probability space Q of all complex functions on J, the Borel field generated by finite dimensional cylinder sets, and a measure P induced by p upon the latter. If p satisfies certain mild differentiability conditions (Loeve, [4]; Doob, [2]), the outer measure P* of C in Q is unity, and both the measure and the stochastic process may be considered as defined on C. Thus the functions of C may be considered as elements of a probability space. Statistical estimation theory has been applied to them (Grenander, [3]; Slepian, [5]). If one takes the values of such a function f(t) at a finite sequence a of points to <t2 < <tnf in J, and forms the expression