Prob. Theory and Math. Stat., pp. 665–684 B. Grigelionis et al. (Eds) © 1999 VSP/TEV LINEAR APPROXIMATION OF RANDOM PROCESSES AND SAMPLING DESIGN PROBLEMS O. V. SELEZNJEV Moscow State University, Faculty of Mathematics and Mechanics, 119 899, Moscow, Russia ABSTRACT Linear approximation of random processes is considered as a three-layers problem: for a single process; for a fixed method, an optimization of a sample points design; for a class of random processes, the best approximation order. The close relationship between the smoothness properties of a function and the best rate of its linear approximation is one of the basic ideas of conventional (deterministic) approximation the- ory. We investigate similar properties for random functions. The Hermite spline interpolation of locally stationary processes and the best approximation order for Hölder’s classes of random processes are stud- ied in more detail. These results can be used, for example, for a justification of a specific approximation method selection in approximation problems. 1. INTRODUCTION 1.1. Motivation Results on approximation of random processes are fundamental for many branches of mathematical statistics (e.g., numerical analysis of random functions, time se- ries analysis, simulation methods, stochastic differential equations). But theoretical achievements in this field are uncomparable with those of the conventional (deter- ministic) approximation theory. Since both stochastic and deterministic methods are used, it causes additional difficulties for an investigation of such problems and for an exposition of results. Even some of these results have been rediscovered. So called Karhunen–Loève method (with a different name) is known and widely used also in classical approximation theory and functional analysis (cf. Schmidt (1907), Ismagilov (1968), Pinkus (1985)). Further, for Gaussian processes, approximation problems in uniform metrics are closely related to extreme value theory (cf. Lead- better, Lindgren, and Rootzén (1983), Piterbarg (1995)). Many problems in time and/or memory consuming computer experiments deal with approximation errors and optimal observation sites (designs). Optimality of designs for approximations is an important subject for environmental and earth sciences, where observations are expensive or unique (see Christakos (1992)). Similar problems arise in archiv- ing and compressing data (realizations of random functions) in databases, especially