Approximation Properties of Chebyshev Polynomials in the Legendre Norm

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.

Automated summary

This paper presents important approximation properties of Chebyshev polynomials in the Legendre norm, focusing on the Chebyshev interpolation operator at the Chebyshev-Gauss-Lobatto points. The results show that the approximation in Legendre norm plays a fundamental role in the numerical analysis of the Legendre-Chebyshev spectral method, and is also useful in Clenshaw-Curtis quadrature based on sampling at Chebyshev points.