Para-orthogonal polynomials in frequency analysis

and we assume α−j = αj, and ω−j = −ωj ∈ (0, π) for j = 1, 2, . . . , I. The constants αj represent amplitudes, the quantities ωj are frequencies, and m is discrete time. The frequency analysis problem is to determine the numbers {αj, ωj : j = 1, 2, . . . , I}, and n0 = 2I when values {x(m) : m = 0, 1, . . . , N − 1} (observations) are known. The Wiener-Levinson method, formulated in terms of Szegő polynomials, can briefly be described as follows (the original ideas of the method can be found in [12, 20]). An absolutely continuous measure ψN is defined on [−π, π] (or on the unit circle T through the transformation θ 7→ z = e) by the formula dψN dθ = 1 2π ∣