We investigate amenability and weak amenability of the l1-algebra of polynomial hypergroups. We derive conditions for (weak) amenability adapted to polynomial hypergroups and show that these conditions are often not satisfied. However, for the hypergroup induced by Chebyshev polynomials of the first kind we prove amenability.

In this paper, we study the asymptotics of the discrete Chebyshev polynomials tn(z,N) as the degree grows to infinity. Global asymptotic formulas are obtained as n → ∞, when the ratio of the parameters n/N = c is a constant in the interval (0, 1). Our method is based on a modified version of the Riemann-Hilbert approach first introduced by Deift and Zhou.

Amongst satisfactory techniques for the numerical solution of differential equations, the use of Chebyshev series is often avoided because of the tedious nature of the calculations. A systematic application of the Chebyshev method is given for certain fourth order boundary value problems in which the derivatives have polynomial coefficients. Numerical results for various problems using the Cheb...

We study the orthogonal polynomial expansion on sparse grids for a function of d variables in a weighted L space. A fast algorithm is developed to compute the orthogonal polynomial expansion by combining the fast cosine transform, a fast transform from the Chebyshev orthogonal polynomial basis to the orthogonal polynomial basis for the weighted L space, and a fast algorithm of computing hierarc...

We describe a fast method for the evaluation of an arbitrary high-dimensional multivariate algebraic polynomial in Chebyshev form at the nodes of an arbitrary rank-1 Chebyshev lattice. Our main focus is on conditions on rank-1 Chebyshev lattices allowing for the exact reconstruction of such polynomials from samples along such lattices and we present an algorithm for constructing suitable rank-1...

The Kepler equation for the parameters of an elliptical orbit, E−ε sin(E)=M , is reduced from a transcendental to a polynomial equation by expanding the sine as a series of Chebyshev polynomials. The single real root is found by applying standard polynomial rootfinders and accepting only the polynomial root that lies on the interval predicted by rigorous theoretical bounds. A complete Matlab im...

Robust polynomial rootfinders can be exploited to compute the roots on a real interval of a nonpolynomial function f(x) by the following: (i) expand f as a Chebyshev polynomial series, (ii) convert to a polynomial in ordinary, series-of-powers form, and (iii) apply the polynomial rootfinder. (Complex-valued roots and real roots outside the target interval are discarded.) The expansion is most e...