Approximation Properties of Chebyshev Polynomials in the Legendre Norm

Cuixia Niu,Hua Wu,Huiqing Liao,Heping Ma

doi:10.3390/math9243271

Abstract

In this paper, we present some important approximation properties of Chebyshev polynomials in the Legendre norm. We mainly discuss the Chebyshev interpolation operator at the Chebyshev–Gauss–Lobatto points. The cases of single domain and multidomain for both one dimension and multi-dimensions are considered, respectively. The approximation results in Legendre norm rather than in the Chebyshev weighted norm are given, which play a fundamental role in numerical analysis of the Legendre–Chebyshev spectral method. These results are also useful in Clenshaw–Curtis quadrature which is based on sampling the integrand at Chebyshev points.

Highlights

Orthogonal polynomials are useful in many areas of numerical analysis and are powerful for function approximation, numerical integration and numerical solution of differential and integral equations [1,2]
The core idea of spectral methods is that any nice enough function can be expanded in a series of orthogonal polynomials so that orthogonal polynomials play a fundamental role in spectral methods [3,4,5,6]
The purpose of this paper is to present some essential approximation results related to Chebyshev polynomials in the Legendre norm

Summary

Introduction

Orthogonal polynomials are useful in many areas of numerical analysis and are powerful for function approximation, numerical integration and numerical solution of differential and integral equations [1,2]. In [11,12], the approximation result of the Chebyshev interpolation operator without the Chebyshev weighted norm was first given. The purpose of this paper is to present some essential approximation results related to Chebyshev polynomials in the Legendre norm. In numerical analysis of the Legendre– Chebyshev spectral method, we need to consider the stability and approximation properties of the Chebyshev interpolation operator in the L2-norm rather than in the Chebyshev weighted norm [13]. Some approximation results in the Legendre norm rather than in the Chebyshev weighted norm are given These results serve as preparations for polynomial-based spectral methods.

Preliminaries and Notations

Approximation Properties of Chebyshev Interpolation Operator in One Dimension

Case of Single Domain in Multi-Dimensions

Case of Multidomain in Multi-Dimensions

Numerical Experiments

Conclusions