Approximation Properties of Chebyshev Polynomials in the Legendre Norm

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.

The Application of Chebyshev Polynomial on the Three-Pass Protocol

“…The name of the Three Pass Protocol comes from the three directions that are carried out by the two parties to communicate [7]. In the Three Pass Protocol, privately each party (1) where , see [13][14][15].…”

“…Chebyshev polynomials have wide applications in applied Mathematics, such as Numerical Analysis, Differential Equations, Cryptography and so on, see [11,12]. Chebyshev polynomials satisfy the composition property, which says for any two positive integers and , we have , [11][12][13][14][15].…”