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The purpose of the paper is to provide a characterization of the error of the best polynomial approximation of composite functions in weighted spaces. Such a characterization is essential for the convergence analysis of numerical methods…
We study sufficient conditions on weight functions under which norm approximations by analytic polynomials are possible. The weights we study include radial, non-radial, and angular weights.
A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…
A rational approximation by a ratio of polynomial functions is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non- Lipschitz functions, where polynomial…
We study the polynomial approximation of symmetric multivariate functions and of multi-set functions. Specifically, we consider $f(x_1, \dots, x_N)$, where $x_i \in \mathbb{R}^d$, and $f$ is invariant under permutations of its $N$…
We seek random versions of some classical theorems on complex approximation by polynomials and rational functions, as well as investigate properties of random compact sets in connection to complex approximation.
Power series representations for special functions are computationally satisfactory only in the vicinity of the expansion point. Thus, it is an obvious idea to use instead Pad\'{e} approximants or other rational functions constructed from…
In this article we obtain new irrationality measures for values of functions which belong to a certain class of hypergeometric functions including shifted logarithmic functions, binomial functions and shifted exponential functions. We…
Functions of one or more variables are usually approximated with a basis: a complete, linearly-independent system of functions that spans a suitable function space. The topic of this paper is the numerical approximation of functions using…
We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, to approximate arbitrary functions. We show that the polynomial coefficients in Legendre expansion, therefore the whole series, converge to zero much more…
We give here the final results about the validity of Jackson-type estimates in comonotone approximation of $2\pi$-periodic functions by trigonometric polynomials. For coconvex and the so called co-$q$-monotone, $q>2$, approximations,…
We established exact in order estimates an approximation of the Sobolev classes $W^{\boldsymbol{r}}_{p,\boldsymbol{\alpha}}(\mathbb{T}^d)$ of periodic functions of many variables with a bounded dominating mixed derivative. The approximation…
This paper presents a quadratic formula-based nonlinear representation for a given single-variable function f(x), $-1 \leq x \leq 1$. First, we construct the explicit polynomial coefficient functions a(x), b(x), and c(x) using a…
Transcendental functions, such as exponentials and logarithms, appear in a broad array of computational domains: from simulations in curvilinear coordinates, to interpolation, to machine learning. Unfortunately they are typically expensive…
We use discrete holomorphic polynomials to prove that, given a refining sequence of critical maps of a Riemann surface, any holomorphic function can be approximated by a converging sequence of discrete holomorphic functions.
In this paper, we propose an interpolation formula for periodic functions. This formula can be regarded as an analog of the Sinc approximation, which is an interpolation formula for functions defined on the entire infinite interval.…
We enhance the approximation capabilities of algebraic polynomials by composing them with homeomorphisms. This composition yields families of functions that remain dense in the space of continuous functions, while enabling more accurate…
We investigate deep composite polynomial approximations of continuous but non-differentiable functions with algebraic cusp singularities. The functions in focus consist of finitely many cusp terms of the form $|x-a_j|^{\alpha_j}$ with…
It is a natural question to ask which plurisubharmonic functions admit a 'nice' approximation in the sense of a decreasing equisingular approximation with analytic singularities. For arbitrary toric plurisubharmonic functions, we give a…
In this paper we study dual bases functions in subspaces. These are bases which are dual to functionals on larger linear space. Our goal is construct and derive properties of certain bases obtained from the construction, with primary focus…