# Andriy PrymakUniversity of Manitoba | UMN · Department of Mathematics

Andriy Prymak

PhD

## About

63

Publications

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327

Citations

Citations since 2017

## Publications

Publications (63)

We show that optimal polynomial meshes exist for every convex body in Rd, confirming a conjecture by A. Kroó.

A set $Q$ in $\mathbb{Z}_+^d$ is a lower set if $(k_1,\dots,k_d)\in Q$ implies $(l_1,\dots,l_d)\in Q$ whenever $0\le l_i\le k_i$ for all $i$. We derive new and refine known results regarding the cardinality of the lower sets of size $n$ in $\mathbb{Z}_+^d$. Next we apply these results for universal discretization of the $L_2$-norm of elements from...

We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $\Omega\subset \mathbb{R}^d$. This new modulus of smoothness is defined via finite differences along the directions of coordinate axes, and along a number of tangential direction...

We show that optimal polynomial meshes exist for every convex body in $\mathbb{R}^d$, confirming a conjecture by A. Kroo.

We prove new Bernstein and Markov type inequalities in $L^p$ spaces associated with the normal and the tangential derivatives on the boundary of a general compact $C^\alpha$-domain with $1\leq \alpha\leq 2$. These estimates are also applied to establish Marcinkiewicz type inequalities for discretization of $L^p$ norms of algebraic polynomials on $C...

Let $\chi(\mathbb{E}^n)$ denote the chromatic number of the Euclidean space $\mathbb{E}^n$, i.e., the smallest number of colors that can be used to color $\mathbb{E}^n$ so that no two points unit distance apart are of the same color. We present explicit constructions of colorings of $\mathbb{E}^n$ based on sublattice coloring schemes that establish...

We show that every $3$-dimensional convex body can be covered by $14$ smaller homothetic copies. The previous result was $16$ copies established by Papadoperakis in 1999, while a conjecture by Hadwiger is $8$. We modify Papadoperakis's approach and develop a discretization technique that reduces the problem to verification of feasibility of a numbe...

Bezdek and Kiss showed that existence of origin-symmetric coverings of unit sphere in ${\mathbb {E}}^n$ by at most $2^n$ congruent spherical caps with radius not exceeding $\arccos \sqrt {\frac {n-1}{2n}}$ implies the X -ray conjecture and the illumination conjecture for convex bodies of constant width in ${\mathbb {E}}^n$ , and constructed such co...

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun only recently. In this paper we obtain a conditional theorem for all integral norms \(L_q\), \(1\le q<\infty \), which is an extension of know...

We prove a new Bernstein type inequality in L p L^p spaces associated with the normal and the tangential derivatives on the boundary of a general compact C 2 C^2 -domain. We give two applications: Marcinkiewicz type inequality for discretization of L p L^p norms and positive cubature formulas. Both results are optimal in the sense that the number o...

For an arbitrary planar convex domain, we compute the behavior of Christoffel function up to a constant factor using comparison with other simple reference domains. The lower bound is obtained by constructing an appropriate ellipse contained in the domain, while for the upper bound an appropriate parallelogram containing the domain is constructed....

This paper studies the behavior of the entropy numbers of classes of functions with bounded integral norms from a given finite dimensional linear subspace. Upper bounds of these entropy numbers in the uniform norm are obtained and applied to establish a Marcinkiewicz-type discretization result for these classes.

This paper studies a new Whitney type inequality on a compact domain $\Omega \subset {\mathbb R}^d$ that takes the form $$ \begin{align*} \inf_{Q\in \Pi_{r-1}^d(\mathcal{E})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{\mathcal{E}}^r(f,\mathrm{diam}(\Omega))_p,\ \ r\in {\mathbb N},\ \ 0
where $\omega _{\mathcal {E}}^r(f, t)_p$ denotes the r th order direc...

Let \(H_n\) be the minimal number of smaller homothetic copies of an n-dimensional convex body required to cover the whole body. Equivalently, \(H_n\) can be defined via illumination of the boundary of a convex body by external light sources. The best known upper bound in three-dimensional case is \(H_3\le 16\) and is due to Papadoperakis. We use P...

The main goal of this work is to give constructions of certain spherical coverings in small dimensions. K. Bezdek and Gy. Kiss showed that existence of origin-symmetric coverings of unit sphere in $\mathbb{E}^n$ by at most $2^n$ congruent spherical caps with radius not exceeding $\arccos\sqrt{\frac{n-1}{2n}}$ implies the X-ray conjecture and the il...

This paper studies a new Whitney type inequality on a compact domain $\Omega\subset {\mathbb{R}}^d$ that takes the form $$\inf_{Q\in \Pi_{r-1}^d({\mathcal{E}})} \|f-Q\|_p \leq C(p,r,\Omega) \omega_{{\mathcal{E}}}^r(f,{\rm diam}(\Omega))_p,\ \ r\in {\mathbb{N}},\ \ 0<p\leq \infty,$$ where $\omega_{{\mathcal{E}}}^r(f, t)_p$ denotes the $r$-th order d...

We prove a new Bernstein type inequality in $L^p$ spaces associated with the tangential derivatives on the boundary of a general compact $C^2$-domain. We give two applications: Marcinkiewicz type inequality for discretization of $L^p$ norm and positive cubature formula. Both results are optimal in the sense that the number of function samples used...

For arbitrary planar convex domain, we compute the behavior of Christoffel function up to a constant factor using comparison with other simple reference domains. The lower bound is obtained by constructing an appropriate ellipse contained in the domain, while for the upper bound an appropriate parallelepiped containing the domain is constructed. As...

This paper studies the behavior of the entropy numbers of classes of functions with bounded integral norms from a given finite dimensional linear subspace. Upper bounds of these entropy numbers in the uniform norm are obtained and applied to establish a Marcinkiewicz type discretization theorem for integral norms of functions from a given finite di...

The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. Even though this problem is extremely important in applications, its systematic study has begun recently. In this paper we obtain a conditional theorem for all integral norms $L_q$, $1\le q<\infty$, which is an extension of known results...

We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact $C^2$-domain $\Omega\subset \mathbb{R}^d$. This new modulus of smoothness is defined via finite differences along the directions of coordinate axes, and along a number of tangential direction...

We prove a general lower bound on Christoffel function on planar convex domains in terms of a modification of the parallel section function of the domain. For a certain class of planar convex domains, in combination with a recent general upper bound, this allows to compute the pointwise behavior of Christoffel function. We illustrate this approach...

The problem is discussed of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure. This problem is investigated for elements of finite-dimensional spaces. Also, discretization of the uniform norm of functions in a given finite-dimensional subspace of continuous f...

Up to a constant factor, we compute the Christoffel function on planar domains with boundary consisting of finitely many C2 curves such that each corner point of the boundary has interior angle strictly between 0 and π. The resulting formula uses the distances from the point of interest to the curves or certain parts of the curves defining the boun...

В статье обсуждается задача о замене интегральной нормы по заданной вероятностной мере соответствующей интегральной нормой по дискретной мере. Указанная задача изучается для элементов конечномерных пространств. Также рассматривается дискретизация равномерной нормы для функций из заданного конечномерного подпространства непрерывных функций. Особое в...

Let H_n be the minimal number of smaller homothetic copies of an n-dimensional convex body required to cover the whole body. Equivalently, H_n can be defined via illumination of the boundary of a convex body by external light sources. The best known upper bound in three-dimensional case is H_3 ≤ 16 and is due to Papadoperakis. We use Papadoperakis'...

Let $H_n$ be the minimal number of smaller homothetic copies of an $n$-dimensional convex body required to cover the whole body. Equivalently, $H_n$ can be defined via illumination of the boundary of a convex body by external light sources. The best known upper bound in three-dimensional case is $H_3\le 16$ and is due to Papadoperakis. We use Papad...

Upper estimates of the diameter and the radius of the family of planar convex bodies with respect to the Banach–Mazur distance are obtained. Namely, it is shown that the diameter does not exceed 19-734≈2.614\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepack...

We compute up to a constant factor the Christoffel function on planar domains with boundary consisting of finitely many $C^2$ curves intersecting at angles which are strictly between $0$ and $\pi$. The resulting formula uses the distances from the point of interest to certain extensions of the curves defining the boundary of the domain.

The problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure is discussed in the paper. The above problem is studied for elements of finite dimensional spaces. Both new results and a survey of known results are presented.

We prove that there is no strongly regular graph (SRG) with parameters (460, 153, 32, 60). The proof is based on a recent lower bound on the number of 4-cliques in a SRG and some applications of Euclidean representation of SRGs.

We prove a general lower bound on Christoffel function on planar convex domains in terms of a modification of the parallel section function of the domain. For a certain class of planar convex domains, in combination with a recent general upper bound, this allows to compute the pointwise behavior of Christoffel function. We illustrate this approach...

We prove the non-existence of strongly regular graph with parameters . We use Euclidean representation of a strongly regular graph together with a new lower bound on the number of 4-cliques to derive strong structural properties of the graph, and then use these properties to show that the graph cannot exist.

New upper bounds on the pointwise behaviour of Christoffel function on convex domains in ${\mathbb{R}}^d$ are obtained. These estimates are established by explicitly constructing the corresponding "needle"-like algebraic polynomials having small integral norm on the domain, and are stated in terms of few easy-to-measure geometric characteristics of...

Nikol’skii inequalities for various sets of functions, domains, and weights will be discussed. Much of the work is dedicated to the class of algebraic polynomials of total degree n on a bounded convex domain D. That is, we study (Formula presented.) for which (Formula presented.)where P is a polynomial of total degree n. We use geometric properties...

For each $q\in{\mathbb{N}}_0$, we construct positive linear polynomial approximation operators $M_n$ that simultaneously preserve $k$-monotonicity for all $0\leq k\leq q$ and yield the estimate \[ |f(x)-M_n(f, x)| \leq c \omega_2^{\varphi^\lambda} \left(f, n^{-1} \varphi^{1-\lambda/2}(x) \left(\varphi(x) + 1/n \right)^{-\lambda/2} \right) , \] for...

We prove that there is no strongly regular graph (SRG) with parameters
(460,153,32,60). The proof is based on a recent lower bound on the number of
4-cliques in a SRG and some applications of Euclidean representation of SRGs.

Our main result is the non-existence of strongly regular graph with
parameters (76,30,8,14). We heavily use Euclidean representation of a strongly
regular graph, and develop a number of tools that allow to establish certain
structural properties of the graph. In particular, we give a new lower bound
for the number of 4-cliques in a strongly regular...

Nikol'skii inequalities for various sets of functions, domains and weights
will be discussed. Much of the work is dedicated to the class of algebraic
polynomials of total degree $n$ on a bounded convex domain $D$. That is, we
study $\sigma:= \sigma(D,d)$ for which \[ \|P\|_{L_q(D)}\le c
n^{\sigma(\frac1p-\frac1q)}\|P\|_{L_p(D)},\quad 0<p\le q\le\in...

We show that for multivariate Freud-type weights
$W_\alpha(x)=\exp(-|x|^\alpha)$, $\alpha>1$, any convex function $f$ on $R^d$
satisfying $fW_\alpha\in L_p(R^d)$ if $1\le p<\infty$, or
$\lim_{|x|\to\infty}f(x)W_\alpha(x)=0$ if $p=\infty$, can be approximated in
the weighted norm by a sequence $P_n$ of algebraic polynomials convex on $R^d$
such that...

We show that on the $d$-dimensional cube $I^d\equiv [0,1]^d$ the discrete
moduli of smoothness which use only the values of the function on a diadic mesh
are sufficient to determine the moduli of smoothness of that function. As an
important special case our result implies for $f\in C(I^d)$ and given integer
$r$ that when $0<\alpha<r$, the condition...

Pippenger (1977) [3] showed the existence of (6m,4m,3m,6)(6m,4m,3m,6)-concentrator for each positive integer mm using a probabilistic method. We generalize his approach and prove existence of (6m,4m,3m,5.05)(6m,4m,3m,5.05)-concentrator (which is no longer regular, but has fewer edges). We apply this result to improve the constant of approximation o...

We prove that for a 3-monotone function F@?C[-1,1], one can achieve the pointwise estimates |F(x)-@J(x)|@?c@w"3(F,@r"n(x)),x@?[-1,1], where @r"n(x)@?1n^2+1-x^2n and c is an absolute constant, both with @J, a 3-monotone quadratic spline on the nth Chebyshev partition, and with @J, a 3-monotone polynomial of degree @?n. The basis for the construction...

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" refers to (finitely many changes of) monotonicity,
convexity, or q-monotonicity of a function (for definition, see Section 4). It
is rather well known that it is possible t...

For a Banach space B of functions which satisfies for some m>0
$$ \max ({\|F+G\|}_B,{\|F-G\|}_B)\geqq ({\|F\|}^s_B+m{\|G\|}^s_B)^{1/s},\quad \forall \,F,G\in B $$
(∗)
a significant improvement for lower estimates of the moduli of smoothness ω
r
(f,t)
B
is achieved. As a result of these estimates, sharp Jackson inequalities which are superior t...

For r≥3, n∈N and each 3-monotone continuous function f on [a,b] (i.e., f is such that its third divided differences [x0,x1,x2,x3]f are nonnegative for all choices of distinct points x0,…,x3 in [a,b]), we construct a spline s of degree r and of minimal defect (i.e., s∈Cr−1[a,b]) with n−1 equidistant knots in (a,b), which is also 3-monotone and satis...

We obtain an analog of Shvedov theorem for convex multivariate approximation by algebras of continuous functions. Comment: 9 pages

For a positive finite measure d μ( u ) on ℝ d normalized to satisfy , the dilated average of f ( x ) is given by
It will be shown that under some mild assumptions on d μ( u ) one has the equivalence
where means , B is a Banach space of functions for which translations are continuous isometries and P ( D ) is an elliptic differential operator indu...

A general approach is given for establishing Nikol'skii-type inequalities for various Lorentz spaces. The key ingredient for the proof is either a Bernstein-type inequality or a Remez-type inequality. Applications of our results to trigonometric polynomials on the torus Td, algebraic polynomials on [- 1,1], spherical harmonic polynomials on the uni...

Given a monotone or convex function on a finite interval we construct splines of arbitrarily high order having maximum smoothness which are "nearly monotone" or "nearly convex" and provide the rate of L p -approximation which can be estimated in terms of the third or fourth (classical or Ditzian–Totik) moduli of smoothness (for uniformly spaced or...

For Ω∈Rd, a convex bounded set with non-empty interior, the moduli of smoothness ωr(f,t)Lq(Ω) and the norm ∥f∥Lq(Ω) are estimated by an Ul’yanov-type expression involving ωr(f,t)Lp(Ω) where 0pq⩽∞. The main result for q∞ is given by ωr(f,t)q⩽C∫0tu-qθωr(f,u)pqduu1/q,0t⩽diamΩ,θ=dp-dq.The inequalities established here settle a conjecture in Ditzian and...

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 piecewise polynomial function (ppf) (or any $q$-monotone
ppf, $q\geq 3$, with one additional degree of smoothness) to be of minimal
defect while keeping it close to the original...

For Lp spaces on Td, Rd and Sd¡1 sharp versions of the classical Marchaud inequality are known. These results are extended here to Orlicz spaces (on Td, Rd and Sd¡1) for which M(u1=q) is convex for some q, 1 < q • 2, where M(u) is the Orlicz function. Sharp converse inequalities for such spaces are deduced.

It is shown that the rate of L p -approximation of a non-decreasing function in L p , 0 < p < ∞, by "nearly non-decreasing" splines can be esti-mated in terms of the third classical modulus of smoothness (for uniformly spaced knots) and third Ditzian-Totik modulus (for Chebyshev knots), and that estimates in terms of higher moduli are impossible. I...

We consider 3-monotone approximation by piecewise polynomials with pre-scribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L1 approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone a...

For every 3-convex piecewise-polynomial function s of degree ≤4 with n equidistant knots on [0, 1] we construct a 3-convex spline s1 (s1 ∈ C(3)) of degree ≤4 with the same knots that satisfies the inequality
$$\left\| {S - S_1 } \right\|_{C_{[0,1]} } \leqslant c\omega _5 (s;1/n),$$
where c is an absolute constant and ω5 is the modulus of smoothnes...

It is proved that for q-convex polynomial approximations, where
$$q \geqslant 4$$
, inequalities of Jackson type are not valid even with a constant depending on the approximated function.