On the inversion of moving averages

Abstract 1. Let an infinite sequence of real numbers be given by (1) {ηt } =[…, η t-2, η t-1, η t , η t+1, η t+2, . . . ] and let (b) = (b 0, b 1,.., bh ) represent real constants. The sequence(2) {ζ t } = […, ζ t-2, ζ t-1, ζt , ζ t+1, ζ t+2,…] defined for every t by the relation(3) ζ t = b 0 · η t + b 1 · η t-1 — . . . + b h-1 · η t-h+1 + b h · η t-h is said to be a moving average of {η t } with weights (b i ) The variable t, which is restricted to integral values 0, ± 1, ± 2 etc., will in the sequel be spoken of as representing time.