Math. Z. 183, 381-398 (1983) Mathematische Zeitschrift 9 Springer-Verlag 1983 An Optimal Spectral Estimator for Multi-Dimensional Time Series with an Infinite Number of Sample Points Palle E.T. Jorgensen Department of Mathematics,Universityof Pennsylvania,Philadelphia, PA 19104, USA 1. Introduction In this note we consider the mathematics of optimal spectral estimators for an unknown deterministic functional trend in a (multi-dimensional) time series model. More specifically, we consider samples s(t)=f(t)+N(t) where the (de- terministic) function f is unknown, and where N(t) is white noise of a given known variance a 2. If t 1, t 2 .... ~]R d are sample points, and /31, f2, ...cC are weights, we consider linear spectral estimators of the form ~ fins(t,). For given n=l a 2, and a finite set of sample points tl,..., tN, the optimal weights/71,..., fN are defined in the literature [4, 9, 16] by a variational principle applied to the mean square error R~. Then R~ is minimized. If the minimum is achieved at 0 0 2 2 0 0 fl,...,fN, then RN=R (fl ..... fiN). The optimal mean square error for an infinite set of sample points {t,},= ~ is then defined conventionally as the limit, 2 _ lira R~. N~o:) In this note we consider a-optimal spectral estimators, and the minimum R 2, for a given infinite set of sample points. By introducing a pair of Hilbert spaces for the variational problem, we observe that R~ can suitably be defined, intrin- sically, for the infinite sample, without having to take the conventional limit N--. c~. We show that the minimum is achieved. The optimal weights are repre- sented by a point f~ in / 2, and we get explicit formulas for the optimal 2 __ weights /3% and Roo-RZ(fl~), which are calculated directly in terms of the infinite sample. (No limit N---,~ is involved!) Assymptotic formulas are ob- tained for the weights and the minimum error, and numerical estimates are calculated in some band-limited models in several dimensions. 2. Mathematical Definitions: The Hilbert Space Formalism Definition 2.1. Let Yr be a complex Hilbert space of tempered distributions on N d, and assume that the Fourier transform f is locally integrable for all fEH.