On least squares discrete Fourier analysis of unequally spaced data

Abstract

. The problem of discrete Fourier analysis of observations at non-equidistant times using the standard set of complex harmonics exp( i 2 πkt ) , t ∈ R , k = 0 , ± 1 , ± 2 , . . . , and the least squares method is studied. The observation model y j = f ( t j )+ η j , j = 1 , . . . , n , is considered for f ∈ L 2 [0 , 1] , where t j ∈ [( j − 1) /n, j/n ) , and η j are correlated complex valued random variables with E η η j = 0 and E η | η j | 2 = σ 2 η < ∞ . Uniqueness and finite sample properties of the observed function Fourier coefficient estimators ˆ c k , k = 0 , ± 1 , . . . , ± m , where m < n/ (8 π ) , obtained by the least squares method, as well as of the corresponding orthogonal projection estimator ˆ f N ( t ) = (cid:80) mk = − m ˆ c k exp( i 2 πkt ) , where N = 2 m + 1 , are examined and compared with those of the standard Discrete Fourier Transform.