SPECTRAL REPRESENTATION AND EXTRAPOLATION OF STATIONARY
RANDOM PROCESSES ON LINEAR SPACES
Lutz Klotz
Manfred Riedel
Abstract: The paper deals with continuous Banach-space-valued stationary random processes
on linear spaces. From von Waldenfels’ [13] integral representation of positive definite
functions on a linear space we derive an analogue of Stone’s theorem for a group of
unitary operators over It is used to obtain spectral representations of a general
Banach-space-valued stationary random process over and its covariance function. For the
special class of Hilbert-Schmidt operator-valued stationary processes the explicit form of
Kolmogorov’s isomorphism theorem between temporal space and spectral space is
established and with its aid there are studied some prediction problems. Our prediction results
are similar to those proved in [5] for multivariate stationary processes on groups.
1991 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -