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Volume 9
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Article

Approximation Properties of Chebyshev Polynomials in the Legendre Norm

by
Cuixia Niu
1,2,
Huiqing Liao
1,
Heping Ma
1,* and
Hua Wu
1
1
Department of Mathematics, Shanghai University, Shanghai 200444, China
2
School of Computer Science and Technology, Shandong Technology and Business University, Yantai 264000, China
*
Author to whom correspondence should be addressed.
Mathematics 2021, 9(24), 3271; https://doi.org/10.3390/math9243271
Submission received: 22 November 2021 / Revised: 10 December 2021 / Accepted: 11 December 2021 / Published: 16 December 2021
(This article belongs to the Special Issue Polynomial Sequences and Their Applications)
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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.
Keywords: Chebyshev polynomials; Chebyshev interpolation operator; the Legendre norm; Legendre–Chebyshev spectral method; Clenshaw–Curtis quadrature; multidomain; multi-dimensions Chebyshev polynomials; Chebyshev interpolation operator; the Legendre norm; Legendre–Chebyshev spectral method; Clenshaw–Curtis quadrature; multidomain; multi-dimensions

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MDPI and ACS Style

Niu, C.; Liao, H.; Ma, H.; Wu, H. Approximation Properties of Chebyshev Polynomials in the Legendre Norm. Mathematics 2021, 9, 3271. https://doi.org/10.3390/math9243271

AMA Style

Niu C, Liao H, Ma H, Wu H. Approximation Properties of Chebyshev Polynomials in the Legendre Norm. Mathematics. 2021; 9(24):3271. https://doi.org/10.3390/math9243271

Chicago/Turabian Style

Niu, Cuixia, Huiqing Liao, Heping Ma, and Hua Wu. 2021. "Approximation Properties of Chebyshev Polynomials in the Legendre Norm" Mathematics 9, no. 24: 3271. https://doi.org/10.3390/math9243271

APA Style

Niu, C., Liao, H., Ma, H., & Wu, H. (2021). Approximation Properties of Chebyshev Polynomials in the Legendre Norm. Mathematics, 9(24), 3271. https://doi.org/10.3390/math9243271

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