APPLICATION OF ORTHOGONALITY ON THE SEMICIRCLE IN SIGNAL PROCESSING

Abstract

In 1987, in Gautschi W., Landau H.J, Milovanović G.V. Polynomials orthogonal on the semicircle, II. Constr. Approx. 1987;3:389–404. the weighted orthogonality on the semicircle and the corresponding orthogonal polynomials were introduced.. Orthogonality is defined using an integral over the area of the upper unit semicircle centered at the origin, without conjugating the second component. Later, the Gaussian quadrature rule on the semicircle was considered. After a long period of time, the authors of this work came up with the idea that quadrature rules on the semicircle could be further developed and improved. The reason for this type of research was the application of these quadrature rules in integrals that frequently appear in signal processing. What differentiates these quadrature rules from the classical ones is the non-Hermitian inner product, which is more advantageous for practical examples, but the validity of standard assertions for quadrature rules must be proven. In this paper, we present applications of all the quadrature formulas on the semicircle that we dealt with and for which we proved the corresponding assertions, primarily in signal processing. Using numerical examples, we illustrate significantly smaller errors when approximately calculating the values of the integrals that appear in signal processing using Gaussian quadrature rules on the semicircle. In addition, we provide quadrature rules through which one can estimate the obtained error with high precision.