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...

In this paper, we develop a novel analytic method to prove the prime number theorem in de la Vall\'ee Poussin's form: $$ \pi(x)=\operatorname{li}(x)+\mathcal O(xe^{-c\sqrt{\log x}}) Instead of performing asymptotic expansion on Chebyshev functions as conventional methods, new approach uses contour-integration analyze Riemann's counting function $J(x)$, which only differs from $\pi(x)$ by $\math...

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.

A method for calculating the strict Chebyshev solution of overdetermined systems of linear equations using linear programming techniques is described. This method provides: (1) a way to determine, for the majority of cases, all the equations belonging to the characteristic set, (2) an efficient method to obtain the inverse of the matrix needed to calculate the strict Chebyshev solution, and (3)...

We analyze whether a given set of analytic functions is an Extended Chebyshev system. This family of functions appears studying the number of limit cycles bifurcating from some nonlinear vector field in the plane. Our approach is mainly based on the so called Derivation-Division algorithm. We prove that under some natural hypotheses our family is an Extended Chebyshev system and when some of th...

We show that viscoelastic effects play a crucial role in the damping of vibrational modes harmonic amorphous solids. The relaxation given plane wave is described by memory function semi-infinite one-dimensions mass-spring chain. initial energy spreads from first site chain to infinity. In beginning chain, there barrier, which significantly reduces decay below Ioffe-Regel frequency. To obtain pa...