Related papers: General Optimal Polynomial Approximants, Stabiliza…
Interpolation and approximation of functionals with conditionally positive definite kernels is considered on sets of centers that are not determining for polynomials. It is shown that polynomial consistency is sufficient in order to define…
Rational best approximations (in a Chebyshev sense) to real functions are characterized by an equioscillating approximation error. Similar results do not hold true for rational best approximations to complex functions in general. In the…
We give a general method to obtain from the integral restrictions of functions sharp pointwise and uniform estimates of these functions. This scheme is illustrated by the examples for Fock\,--\,Bargmann spaces of entire functions of several…
We study an optimization problem in which the objective is given as a sum of logarithmic-polynomial functions. This formulation is motivated by statistical estimation principles such as maximum likelihood estimation, and by loss functions…
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…
Approximation theorists have established best-in-class optimal approximation rates of deep neural networks by utilizing their ability to simultaneously emulate partitions of unity and monomials. Motivated by this, we propose partition of…
A Boolean function $g$ is said to be an optimal predictor for another Boolean function $f$, if it minimizes the probability that $f(X^{n})\neq g(Y^{n})$ among all functions, where $X^{n}$ is uniform over the Hamming cube and $Y^{n}$ is…
This paper studies the polynomial optimization problem whose feasible set is a union of several basic closed semialgebraic sets. We propose a unified hierarchy of Moment-SOS relaxations to solve it globally. Under some assumptions, we prove…
We consider the approximation of a continuous function, defined on a compact set of the $d$-dimensional Euclidean space, by sums of two ridge functions. We obtain a necessary and sufficient condition for such a sum to be a best…
Given a nonnegative polynomial f, we provide an explicit expression for its best $\ell_1$-norm approximation by a sum of squares of given degree.
Let $E$ be an arbitrary subset of the unit circle $T$ and let $f$ be a function defined on $E$. When there exist polynomials $P_n$ which are uniformly bounded by a number $M > 0$ on $T$ and converge (pointwise) to $f$ at each point of $E$?…
The discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces is considered. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm…
We study the computability of the operator norm of a matrix with respect to norms induced by linear operators. Our findings reveal that this problem can be solved exactly in polynomial time in certain situations, and we discuss how it can…
We study worst-case optimal approximation of positive linear functionals in reproducing kernel Hilbert spaces induced by increasingly flat Gaussian kernels. This provides a new perspective and some generalisations to the problem of…
Recent years have witnessed the introduction and development of extremely fast rational function algorithms. Many ideas in this realm arose from polynomial-based linear-algebraic algorithms. However, polynomial approximation is occasionally…
Estimates are obtained for the initial coefficients of a normalized analytic function $f$ in the unit disk $\mathbb{D}$ such that $f$ and the analytic extension of $f^{-1}$ to $\mathbb{D}$ belong to certain subclasses of univalent…
The main object of the present paper is to give a complete result regarding the best approximation rate of certain trigonometric series in general complex valued continuous function space under a new condition which gives an essential…
This paper focuses on the study of a mathematical program with equilibrium constraints, where the objective and the constraint functions are all polynomials. We present a method for finding its global minimizers and global minimum using a…
Sparse polynomial approximation has become indispensable for approximating smooth, high- or infinite-dimensional functions from limited samples. This is a key task in computational science and engineering, e.g., surrogate modelling in…
Generic approximation of entire functions by their Pad\'{e} approximants has been achieved in the past (\cite{3}). In the present article we obtain generic approximation of holomorphic functions on arbitrary open sets by sequences of their…