Orthogonal Polynomials

Abstract

Introduction: Classical orthogonal polynomials such as the Legendre, Laguerre, Hermite, and Chebyshev polynomials have a wide range of applications across various domains of science and engineering.

Aim: The aim of this research paper is to present selected recent developments in the theory of orthogonal polynomials, with particular emphasis on their analytical properties and their role in approximation theory.

Methods: The study employs a combined quantitative-qualitative research methodology. It is based on the analysis of relevant scientific literature, from which data significant to the subject matter were collected, interpreted, and systematically examined.

Results: The results indicate that contemporary research in probability theory, graph theory, coding theory, and related areas increasingly relies on the theory of orthogonal polynomials. This paper provides a theoretical analysis of the connection between orthogonal polynomials and their applications in specific computational problems. Fundamental properties of these polynomials are presented and illustrated through selected examples that contribute to a clearer understanding of their structure and practical relevance.

Conclusion: The approximation of the transfer function of a low-pass filter can be achieved through a straightforward adaptation of orthogonal Jacobi polynomials. Furthermore, Gaussian quadrature represents a powerful numerical technique for the approximation of definite integrals, utilizing optimally chosen nodes and weight functions to achieve high accuracy with minimal computational effort. Orthogonal polynomials thus serve as a significant link between pure mathematics and engineering disciplines, highlighting the importance of further study on their properties and applications.