Related papers: The error of Chebyshev approximations on shrinking…
We address the problem of the best uniform approximation by linear combinations of a finite system of functions. If the system is Chebyshev and the problem is unconstrained, then the classical Remez algorithm provides a fast and precise…
The challenge to measure exposures regularly forces financial institutions into a choice between an overwhelming computational burden or oversimplification of risk. To resolve this unsettling dilemma, we systematically investigate replacing…
The theory of Chebyshev (uniform) approximation for univariate polynomial and piecewise polynomial functions has been studied for decades. The optimality conditions are based on the notion of alternating sequence. However, the extension the…
We discuss the relation between the linear Tschebyshev-Pad\'e approximations to analytic function $f$ and the diagonal type I Hermite-Pad\'e polynomials for the tuple of functions $[1,f_1,f_2]$ where the pair of functions $f_1,f_2$ forms…
For trigonometric series and series of Chebyshev polynomials, we defined trigonometric Hermite-Pad\'e and Hermite-Jacobi approximations, linear and nonlinear Hermite-Chebyshev approximations. We established criterion of the existence and…
This paper considers the approximation of a monomial $x^n$ over the interval $[-1,1]$ by a lower-degree polynomial. This polynomial approximation can be easily computed analytically and is obtained by truncating the analytical Chebyshev…
We derive optimal asymptotic and non-asymptotic lower bounds on the Widom factors for weighted Chebyshev and orthogonal polynomials on compact subsets of the real line. In the Chebyshev case we extend the optimal non-asymptotic lower bound…
For a function that is analytic on and around an interval, Chebyshev polynomial interpolation provides spectral convergence. However, if the function has a singularity close to the interval, the rate of convergence is near one. In these…
The present paper concerns filtered de la Vall\'ee Poussin (VP) interpolation at the Chebyshev nodes of the four kinds. This approximation model is interesting for applications because it combines the advantages of the classical Lagrange…
The Chisholm rational approximant is a natural generalization to two variables of the well-known single variable Pad\'e approximant, and has the advantage of reducing to the latter when one of the variables is set equals to 0. We present,…
Some new results on the convergence of nonlinear diagonal Pad\'e--Chebyshev approximations to multivalued analytic function given on the segment $[-1,1]$, are proved. We show that these approximations converge to the given function in the…
Estimating the Domain of Attraction (DA) of non-polynomial systems is a challenging problem. Taylor expansion is widely adopted for transforming a nonlinear analytic function into a polynomial function, but the performance of Taylor…
This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform…
A method is suggested for interpolating between small-variable and large-variable asymptotic expansions. The method is based on self-similar approximation theory resulting in self-similar root approximants. The latter are more general than…
Adaptive approximation (or interpolation) takes into account local variations in the behavior of the given function, adjusts the approximant depending on it, and hence yields the smaller error of approximation. The question of constructing…
We present a new rational approximation algorithm based on the empirical interpolation method for interpolating a family of parametrized functions to rational polynomials with invariant poles, leading to efficient numerical algorithms for…
Consider a rational family of planar rational curves in a certain region of interest. We are interested in finding an approximation to the implicit representation of the envelope. Since exact implicitization methods tend to be very costly,…
We present numerical solutions for differential equations by expanding the unknown function in terms of Chebyshev polynomials and solving a system of linear equations directly for the values of the function at the extrema (or zeros) of the…
As one myth of polynomial interpolation and quadrature, Trefethen [30] revealed that the Chebyshev interpolation of $|x-a|$ (with $|a|<1 $) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation…
We study diagonal multipoint Pad\'e approximants to sums of a Cauchy transform of a complex measure and a rational function. The measure is assumed to have compact regular support included into the real line and an argument of bounded…