An application of Szegő quadratures to the computation of the Fourier transform (Q884105)

The computation of the Fourier transform \[ {\mathcal F}(g)(w):= \int^\infty_{-\infty} g(t) e^{2\pi iwt}\,dt,\quad -\infty< w<\infty, \] is of considerable difficulty involving the infinite range and the rapidly oscillatory integrand. To overcome these computational problems, in the present paper Szegő quadrature formulas are revised. Szegő quadratures (exactly integrating trigonometric polynomials of degree as high as possible) represent the analogue on the unit circle of the well-known Gauss-Christoffel quadratures for intervals on the real line. As to the theoretical background, the reader is referred to the paper by \textit{W. B. Jones, O. Njåstad} and \textit{W. J. Thron} [Bull. Lond. Math. Soc. 21, No. 2, 113--152 (1989; Zbl 0637.30035)]. The present paper lays special emphasis on numerical methods to compute the Fourier transform, which is of growing interest in the fields of digital signal processing, operator theory, and probability theory.