Gauss and Clenshaw–Curtis quadrature, like Legendre and Chebyshev spectral methods, make use of grids strongly clustered at boundaries. From the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal. Nevertheless such methods may “waste” a factor of π/2 with respect to each space dimension. We propose new nonpolynomial quadrature methods that avoid th...

Let T be a Jordan curve in the complex plane, and let Í) be the compact set bounded by T. Let / denote a function analytic on O. We consider the approximation of / on fî by a polynomial p of degree less than n that interpolates / in n points on T. A convenient way to compute such a polynomial is provided by the Newton interpolation formula. This formula allows the addition of one interpolation ...

Gauss and Clenshaw–Curtis quadrature, like Legendre and Chebyshev spectral methods, make use of grids strongly clustered at boundaries. From the viewpoint of polynomial approximation this seems necessary and indeed in certain respects optimal. Nevertheless such methods may “waste” a factor of π/2 with respect to each space dimension. We propose new nonpolynomial quadrature methods that avoid th...

In this paper, we give a unified approach to error estimates for interpolation on sparse Gauß–Chebyshev grids for multivariate functions from Besov–type spaces with dominating mixed smoothness properties. The error bounds obtained for this method are almost optimal for the considered scale of function spaces. 1991 Mathematics Subject Classification: 41A05, 41A63, 65D05, 46E35

This paper provides a rigorous and delicate analysis for exponential decay of Jacobi polynomial expansions of analytic functions associated with the Bernstein ellipse. Using an argument that can recover the best estimate for the Chebyshev expansion, we derive various new and sharp bounds of the expansion coefficients, which are featured with explicit dependence of all related parameters and val...

In this paper, an Adomian decomposition method using Chebyshev orthogonal polynomials is proposed to solve a well-known class of weakly singular Volterra integral equations. Comparison with the collocation method using polynomial spline approximation with Legendre Radau points reveals that the Adomian decomposition method using Chebyshev orthogonal polynomials is of high accuracy and reduces th...

On the line and its tensor products, Fekete points are known to be the Gauss–Lobatto quadrature points. But unlike high-order quadrature, Fekete points generalize to non-tensor-product domains such as the triangle. Thus Fekete points might serve as an alternative to the Gauss–Lobatto points for certain applications. In this work we present a new algorithm to compute Fekete points and give resul...