On Linear Estimation Theory for an Infinite Number of Observations

We consider a stochastic process, subject to the condition that it be representable as a linear combination of a finite number of given functions (the coefficients of the linear combination are assumed to be independent). Among the linear functionals of the stochastic process it is required to find the best unbiased estimate for the linear form of the independent coefficients. The existence of such an estimate is established in Theorem 3.1. The results obtained are natural generalizations of the classical method of least squares to the case of Hilbert space. The given problem can also be considered as a generalization of the well-known problem of Zadeh and Ragazzini [2] on the estimation of a polynomial form against a background of a stationary signal and stationary noise.