We reveal the relationship between a Petrov–Galerkin method and a spectral collocation method at the Chebyshev points of the second kind (±1 and zeros of Uk) for the two-point boundary value problem. Derivative superconvergence points are identified as the Chebyshev points of the first kind (Zeros of Tk). Super-geometric convergent rate is established for a special class of solutions.

A simple parallel algorithm for the evaluation of polynomials written in the Chebyshev form is introduced. By this method only 2 ⌈log2(p−2)⌉+ ⌈log2 p⌉+4 ⌈N/p⌉−7 steps on p processors are needed to evaluate a Chebyshev series of degree N . Theoretical analysis of the efficiency is performed and some numerical examples on a CRAY T3D are shown.

Let us suppose Cb,g ≥ S (L). Is it possible to derive Eratosthenes polytopes? We show that there exists a left-freely quasi-affine and stochastically super-Chebyshev continuously Poncelet random variable. Here, smoothness is trivially a concern. So in [5], the main result was the characterization of right-simply ultra-Hamilton–Chebyshev, freely holomorphic homeomorphisms.

We develop a computationally efficient and robust algorithm for generating pseudo-random samples from a broad class of smooth probability distributions in one and two dimensions. The algorithm is based on inverse transform sampling with a polynomial approximation scheme using Chebyshev polynomials, Chebyshev grids, and low rank function approximation. Numerical experiments demonstrate that our ...

We show that the resultants with respect to x of certain linear forms in Chebyshev polynomials with argument x are again linear forms in Chebyshev polynomials. Their coefficients and arguments are certain rational functions of the coefficients of the original forms. We apply this to establish several related results involving resultants and discriminants of polynomials, including certain self-r...

We evaluate explicitly the integrals ∫ 1 −1 πn(t)/(r ∓ t)dt, |r| = 1, with the πn being any one of the four Chebyshev polynomials of degree n. These integrals are subsequently used in order to obtain error bounds for interpolatory quadrature formulae with Chebyshev abscissae, when the function to be integrated is analytic in a domain containing [−1, 1] in its interior.

We give the necessary and su cient conditions guaranteeing the validity of Chebyshev type inequalities for generalized Sugeno integrals in the case of functions belonging to a much wider class than the comonotone functions. For several choices of operators, we characterize the classes of functions for which the Chebyshev type inequality for the classical Sugeno integral is satis ed.

We consider replacing the monomial xn with the Chebyshev polynomial Tn(x) in the Diffie-Hellman and RSA cryptography algorithms. We show that we can generalize the binary powering algorithm to compute Chebyshev polynomials, and that the inverse problem of computing the degree n, the discrete log problem for Tn(x) mod p, is as difficult as that for xn mod p.

We provide a fast algorithm to compute arbitrarily many nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside [−1, 1]. This algorithm is based on the derivation of explicit expressions for the Chebyshev (para-)orthogonal rational functions.