Continued fractions and Szego polynomials in frequency analysis and related topics

This paper is an expository survey of recent research on the application of Szego polynomials and PPC-continued fractions to the frequency analysis pr oblem described as follows: We want to determine the unknown frequencies om ega(1), omega(2), ..., omega(I) from a sample of N observed values x(N)(m), m = 0, 1, ..., N - 1, arising from a continuous waveform that is the super position of a finite number of sinusoidal waves with frequencies omega(1), omega(2), ..., omega(I). The method is based on the property that certain z eros of the Szego polynomials (and poles of the PPC-fraction approximants) converge (as N --> infinity) to the frequency points e(i omega j), j = +/- 1, +/- 2, ..., +/- I. The remaining zeros are bounded away from the unit ci rcle \z\ = 1, as N --> infinity. The Levinson algorithm is used to construc t the Szego polynomials and PPC-fractions from the values x(N)(m). A discus sion is given on connections between the topics: Caratheodory functions, th e trigonometric moment problem, Szego polynomials and PPC-fractions. We als o describe applications to Doppler radar, medicine, speech processing, spee ch therapy, meteorology and ocean tides.