K. Kopotun

activity

20112023

most citedUniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials

33 citations · 38 across the 14 of their papers we have counts for

collaborators

19 papers

math.CA2023

Onesided, intertwining, positive and copositive polynomial approximation with interpolatory constraints

German Dzyubenko, Kirill A. Kopotun

Given $k \in N$ , a nonnegative function $f \in C^{r} [a, b]$ , $r \geq 0$ , an arbitrary finite collection of points ${α_{i}}_{i \in J} \subset [a, b]$ , and a corresponding collection…

math.CA2020

Exact pointwise estimates for polynomial approximation with Hermite interpolation

Kirill A. Kopotun, Dany Leviatan, Igor A. Shevchuk

We establish best possible pointwise (up to a constant multiple) estimates for approximation, on a finite interval, by polynomials that satisfy finitely many (Hermite) interpolatio…

math.CA2020

Interpolatory pointwise estimates for convex polynomial approximation

K. A. Kopotun, D. Leviatan, I. Petrova +1

This paper deals with approximation of smooth convex functions $f$ on an interval by convex algebraic polynomials which interpolate $f$ at the endpoints of this interval. We call s…

math.CA2019★ 1 cited

Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update

K. A. Kopotun, D. Leviatan, I. A. Shevchuk

It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it tur…

math.CA2019

On one estimate of divided differences and its applications

K. A. Kopotun, D. Leviatan, I. A. Shevchuk

We give an estimate of the general divided differences $[x_{0}, \dots, x_{m}; f]$ , where some of the $x_{i}$ 's are allowed to coalesce (in which case, $f$ is assumed to be sufficiently smoo…

math.CA2019

On some properties of moduli of smoothness with Jacobi weights

K. A. Kopotun, D. Leviatan, I. A. Shevchuk

We discuss some properties of the moduli of smoothness with Jacobi weights that we have recently introduced and that are defined as \[ ω_{k,r}^φ(f^{(r)},t)_{α,β,p} :=\sup_{0\leq h\…

math.CA2018

Interpolatory estimates for convex piecewise polynomial approximation

Kirill A. Kopotun, Dany Leviatan, Igor A. Shevchuk

In this paper, among other things, we show that, given $r \in N$ , there is a constant $c = c (r)$ such that if $f \in C^{r} [- 1, 1]$ is convex, then there is a number ${\mathcal N}={\mathca…

math.CA2017

Interpolatory pointwise estimates for monotone polynomial approximation

K. A. Kopotun, D. Leviatan, I. A. Shevchuk

Given a nondecreasing function $f$ on $[- 1, 1]$ , we investigate how well it can be approximated by nondecreasing algebraic polynomials that interpolate it at $\pm 1$ . We establish p…

math.CA2017

On weighted approximation with Jacobi weights

Kirill A. Kopotun, Dany Leviatan, Igor A. Shevchuk

We obtain matching direct and inverse theorems for the degree of weighted $L_{p}$ -approximation by polynomials with the Jacobi weights $(1 - x)^{α} (1 + x)^{β}$ . Combined, the estimates yield…

math.CA2017

On moduli of smoothness with Jacobi weights

Kirill A. Kopotun, Dany Leviatan, Igor A. Shevchuk

The main purpose of this paper is to introduce moduli of smoothness with Jacobi weights $(1 - x)^{α} (1 + x)^{β}$ for functions in the Jacobi weighted $L_{p} [- 1, 1]$ , $0 < p \leq \infty$ , spaces.…

math.CA2016★ 1 cited

Yet another look at positive linear operators, $q$ -monotonicity and applications

K. Kopotun, D. Leviatan, A. Prymak +1

For each $q \in N_{0}$ , we construct positive linear polynomial approximation operators $M_{n}$ that simultaneously preserve $k$ -monotonicity for all $0 \leq k \leq q$ and yield…

math.CA2015

Uniform polynomial approximation with $A^{*}$ weights having finitely many zeros

Kirill A. Kopotun

We prove matching direct and inverse theorems for uniform polynomial approximation with $A^{*}$ weights (a subclass of doubling weights suitable for approximation in the $L_{\infty}$ n…

math.CA2014

New moduli of smoothness

K. A. Kopotun, D. Leviatan, I. A. Shevchuk

In this paper, we discuss various properties of the new modulus of smoothness \[ ω^φ_{k,r}(f^{(r)},t)_p := \sup_{0 < h\leq t}\|\mathcal W^r_{kh}(\cdot) Δ_{hφ(\cdot)}^k (f^{(r)},\cd…

math.CA2014

Polynomial approximation with doubling weights having finitely many zeros and singularities

Kirill A. Kopotun

We prove matching direct and inverse theorems for (algebraic) polynomial approximation with doubling weights $w$ having finitely many zeros and singularities (i.e., points where $w…

math.CA2014

New moduli of smoothness: weighted DT moduli revisited and applied

K. A. Kopotun, D. Leviatan, I. A. Shevchuk

We introduce new moduli of smoothness for functions $f \in L_{p} [- 1, 1] \cap C^{r - 1} (- 1, 1)$ , $1 \leq p \leq \infty$ , $r \geq 1$ , that have an $(r - 1)$ st locally absolutely continuous derivative…

math.CA2014

Weighted moduli of smoothness of $k$ -monotone functions and applications

Kirill A. Kopotun

Let $ω_{φ}^{k} (f, δ)_{w, L_{q}}$ be the Ditzian-Totik modulus with weight $w$ , $M^{k}$ be the cone of $k$ -monotone functions on $(- 1, 1)$ , i.e., those functions whose $k$ th divided difference…

math.CA2014

Polynomial approximation with doubling weights

Kirill A. Kopotun

Among other things, we prove that, for a doubling weight $w$ , $0 < p \leq \infty$ , $r \in N_{0}$ , and $0 < α < r + 1 - 1/ λ_{p}$ , we have \[ E_n(f)_{p, w_n} = O(n^{-α}) \iff ω_φ^{r+1}(f,…

math.NA2014★ 3 cited

Constrained Spline Smoothing

K. Kopotun, D. Leviatan, A. Prymak

Several results on constrained spline smoothing are obtained. In particular, we establish a general result, showing how one can constructively smooth any monotone or convex piecewi…

math.CA2011★ 33 cited

Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials

K. A. Kopotun, D. Leviatan, A. Prymak +1

We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a finite interval. In this article, "shape" re…

Onesided, intertwining, positive and copositive polynomial approximation with interpolatory constraints

Exact pointwise estimates for polynomial approximation with Hermite interpolation

Interpolatory pointwise estimates for convex polynomial approximation

Uniform and pointwise shape preserving approximation (SPA) by algebraic polynomials: an update

On one estimate of divided differences and its applications

On some properties of moduli of smoothness with Jacobi weights

Interpolatory estimates for convex piecewise polynomial approximation

Interpolatory pointwise estimates for monotone polynomial approximation

On weighted approximation with Jacobi weights

On moduli of smoothness with Jacobi weights

Yet another look at positive linear operators, q-monotonicity and applications

Uniform polynomial approximation with A∗ weights having finitely many zeros

New moduli of smoothness

Polynomial approximation with doubling weights having finitely many zeros and singularities

New moduli of smoothness: weighted DT moduli revisited and applied

Weighted moduli of smoothness of k-monotone functions and applications

Polynomial approximation with doubling weights

Constrained Spline Smoothing

Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials

Yet another look at positive linear operators, $q$ -monotonicity and applications

Uniform polynomial approximation with $A^{*}$ weights having finitely many zeros

Weighted moduli of smoothness of $k$ -monotone functions and applications