Szegő polynomials applied to frequency analysis
Journal of Computational and Applied Mathematics - Special issue on computational complex analysis
Asymptotics for zeros of Szegő polynomials associated with trigonometric polynomial signals
Journal of Approximation Theory
Modification of a method using Szeg&ohuml; polynomials in frequency analysis: the V-process
Journal of Computational and Applied Mathematics - Special issue on orthogonal polynomials, special functions and their applications
Extrapolation, Interpolation, and Smoothing of Stationary Time Series
Extrapolation, Interpolation, and Smoothing of Stationary Time Series
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Asymptotic behavior of Szegö polynomials
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the conference on orthogonal functions and related topics held in honor of Olav Njåstad
Asymptotic behavior of Szegö polynomials
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the conference on orthogonal functions and related topics held in honor of Olav Njåstad
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For a trigonometric signal an important problem is to find the unknown frequencies from a set of observations. This is called the frequency analysis problem.One method is to construct, from measured values, a family of positive measures, depending upon one or more parameters. These define in turn an inner product, positive definite sequences of moments, and sequences of polynomials (or more generally rational functions), orthogonal on the unit circle. Asymptotic properties of the zeros then lead to the frequencies.In recent variations of this method the moments are modified. If the new sequence also is positive definite we are back to the method just mentioned. But sometimes it is difficult to find out if it is positive definite or not.A "strange-looking" modification, multiplication of the "moments" by Rn2 where R∈(0,1) was introduced very recently. This modification turned out to be rewarding in different surprising ways in all examples carried out, although at first the question of positive definiteness was unsettled. In the present paper, the positive definiteness is proved. Moreover, it is proved that for α R|n|α does not generally lead to a positive definite sequence.