A result on the existence and uniqueness of metric projection for certain sets is proved, by means of a saddle point theorem. A conjecture, based on such a result and aiming for the construction of a nonconvex Chebyshev set in a Hilbert space, is presented.

A weighted orthogonal system on the half line based on the Chebyshev rational functions is introduced. Basic results on Chebyshev rational approximations of several orthogonal projections and interpolations are established. To illustrate the potential of the Chebyshev rational spectral method, a model problem is considered both theoretically and numerically: error estimates for the Chebyshev ra...

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...

We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions. Keywords—Approximation Theory, Chebyshev Polynomial, Computable Functions, Comp...

We introduce a new and efficient Chebyshev-Legendre Galerkin method for elliptic problems. The new method is based on a Legendre-Galerkin formulation, but only the Chebyshev-Gauss-Lobatto points are used in the computation. Hence, it enjoys advantages of both the LegendreGalerkin and Chebyshev-Galerkin methods.

We give two recursive expressions for both MacWilliams and Chebyshev matrices. The expressions give rise to simple recursive algorithms for constructing the matrices. In order to derive the second recursion for the Chebyshev matrices we find out the Krawtchouk coefficients of the Discrete Chebyshev polynomials, a task interesting on its own.

Nodal point sets, and associated collocation projections, play an important role in a range of high-order methods, including Flux Reconstruction (FR) schemes. Historically, efforts have focused on identifying nodal sets that aim to minimise the L∞ error interpolating polynomial. The present work combines comprehensive review known approximation theory results, with new numerical experiments, mo...

We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains (e.g, the Cantor set). In particular, we present a new expression for the discrete Gaussian curvature which turns out to be valid for asymptotic nets on any time sca...