Evolutionary Spectra and Non-Stationary Processes

First it is shown how the Karhunen-Loeve approach to the detection of a deterministic signal can be given a coordinate-free and geometric interpretation in a particular Hilbert space of functions that is uniquely determined by the covariance function of the additive Gaussian noise. This Hilbert space, which is called a reproducing-kernel Hilbert space (RKHS), has many special properties that appear to make it a natural space of functions to associate with a second-order random process. A mapping between the RKHS and the linear Hilbert space of random variables generated by the random process is studied in some detail. This mapping enables one to give a geometric treatment of the detection problem. The relations to the usual integral-equation approach to this problem are also discussed. Some of the special properties of the RKHS are developed and then used to study the singularity and stability of the detection problem and also to suggest simple means of approximating the detectability of the signal. The RKHS for several multidimensional and multivariable processes is presented; by going to the RKHS of functionals rather than functions it is also shown how generalized random processes, including white noise and stationary processes whose spectra grow at infinity, are treated.

RKHS approach to detection and estimation problems--I: Deterministic signals in Gaussian noise

The complex moment problem and subnormality : a polar decomposition approach

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Quasi-analytic vectors

I n a beaut i fu l p a p e r [13], 1~I. Kre in has given a pene t r a t i ng analys is of a general p rob lem of momen t s (see also [17]). This p a p e r is the cu lmina t ion of a series of notes and papers b y K r e i n s t re tch ing over a lmos t a decade (see [13] for a bibl iog raphy) . His resul ts a p p e a r in the form of a t h e o r y a b o u t a special class of symmet r i c opera to r s on a Hi ]ber t space (see also [16]). The p r o t o t y p e of an opera to r in th is class m a y be found in the t h e o r y of the classical H a m b u r g e r m o m e n t problem. The genera l p rob lem of momen t s which can be t r e a t e d b y K re in ' s me thods is concerned wi th condi t ions on a p re -Hi ]be r t space E of ana ly t i c funct ions of a single real va r iab le for which there exis ts a measure d/~ (t)~>0 so t h a t if /, g E C, then

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On the extensions of positive definite functions

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A. Devinatz

Papers

Integral representations of positive definite functions. ii