Comparative Analysis of Least squares Approximation Using Shifted Legendre and Hermite Polynomials

Abstract

This study presents a comparative analysis of least squares approximation techniques utilizing shifted Legendre and Hermite  polynomials. Both polynomial families are integral to numerical analysis and function approximation, yet they exhibit distinct  characteristics that impact their performance in least squares fitting. Shifted Legendre polynomials, defined over a finite interval, provide  optimal properties for approximating functions within bounded domains, while Hermite polynomials, characterized by their orthogonality and exponential decay, are particularly suited for problems involving infinite intervals or rapid decay functions. We  investigate the accuracy and efficiency of least squares approximations using these polynomial bases across various test functions,  including smooth, discontinuous, and oscillatory functions. The findings reveal that while shifted Legendre polynomials excel in  approximating functions within a bounded range, Hermite polynomials demonstrate superior performance for functions defined over infinite intervals or with specific decay properties. This study provides insights into the selection criteria for polynomial bases in least  squares approximation, facilitating better decisions in practical applications such as data fitting, numerical integration, and solving  differential equations.