András Kroó

The density of extreme points in complex polynomial approximation

Let K be a compact set in the complex plane having connected and regular complement, and let f be any function continuous on K and analytic in the interior of K. For the polynomials p* (f) of respective degrees at most n of best uniform approximation to f on K, we investigate the density of the sets of extreme points An(f):= {z E K: If (z))-Pn(f)(Z)I = Ilf -Pn*(f)IIK } in the boundary of K.

On density of interpolation points, a Kadec-type theorem, and Saff's principle of contamination inLp-approximation

Letf ∈Lp(I) and denote byBn,p(f) the polynomial of bestLp-approximation tof of degreen (1<p<∞,I=[−1,1], the norm is weightedLp-norm with an arbitrary positive weight). Extending a result proved by Saff and Shekhtman forp=2 we show that for every 1<p<∞ andf ∈Lp(I) (not a polynomial) points of sign change of the error functionf-Bn,p(f) are dense inI asn→∞.

A Sharp Version of Mahler's Inequality for Products of Polynomials

In this note we give some sharp estimates for norms of polynomials via the products of norms of their linear terms. Different convex norms on the unit disc are considered.

On the uniform modulus of continuity of the operator of best approximation in the space of periodic functions

Introducion and preliminary results In this paper we shall present some results connected with the uniform continuity of best approximations. In the last fifteen years problems dealing with local continuity of best approximation have been widely investigated. It was proved ([1], [2]) that the metric projection operator onto a finite-dimensional ~ebysev subspace of C[a, b] is pointwise Lip 1. An analogous result was obtained for the space Lp, 2 0 and Me= C[a, b] we can introduce the uniform modulus of continuity of the operator of best approximation on M as

A Weierstrass-type theorem for homogeneous polynomials

By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in R d . In this paper we study the following question: does the density hold if we approximate only by homogeneous polynomials? Since the set of homogeneous polynomials is nonlinear this leads to a nontrivial problem. It is easy to see that: 1) density may hold only on star-like 0-symmetric surfaces; 2) at least 2 homogeneous polynomials are needed for approximation. The most interesting special case of a star-like surface is a convex surface. It has been conjectured by the second author that functions continuous on 0-symmetric convex surfaces in R d can be approximated by a pair of homogeneous polynomials. This conjecture is not resolved yet but we make substantial progress towards its positive settlement. In particular, it is shown in the present paper that the above conjecture holds for 1) d = 2, 2) convex surfaces in R d with C 1+ǫ boundary.

Interpolatory properties of best rationalL1-approximations

In this paper the distribution of the zeros of the error function for bestL1-approximation by rational functions fromRn,m is considered. It is shown that the maximal distance between such zeros isO(1/(n−m)), ifn > m.

Chebyshev rank in ₁-approximation

Caracterisation des sous-espaces de fonctions continues ayant un rang de Chebyshev borne par rapport a des poids positifs bornes. Applications diverses

A comparison of uniform and discrete polynomial approximation

AbstractИжУЧАЕтсь скОРОсть с хОДИМОстИ тРИгОНОМЕ тРИЧЕскИх пОлИНОМОВ НАИлУЧшЕг О РАВНО-МЕРНОгО пРИБлИжЕНИь пЕРИОДИ ЧЕскОИ ФУНкцИИ НА кОН ЕЧНОМ МНОжЕстВЕ тОЧЕк к пОл ИНОМУ НАИлУЧшЕгО РАВНОМЕР НОгО пРИБлИжЕНИь ФУН кцИИ НА ВсЕМ пЕРИОДЕ, кОгДА ДИАМЕт Р МНОжЕстВА стРЕМИтсь к НУлУ. ДАЕт сь тАкжЕ ОцЕНкА ОтклО НЕНИь ВЕлИЧИН НАИлУЧшЕгО п РИБлИжЕНИь НА ВсЕМ пЕРИОДЕ И НА кОНЕ ЧНОМ МНОжЕстВЕ тОЧЕк И ИсслЕДУЕтсь сВьжь с Д ИФФЕРЕНцИАльНыМИ сВОИстВАМИ пРИБлИжА ЕМых ФУНкцИИ.

On Remez-type inequalities for polynomials in Rm and Cm

Remez-type inequalities provide estimates for the size of polynomials on given sets K⊂Rm (or Cm) when the magnitude of polynomials on “largeldquo” subsets of K is known. We shall study this question on “smooth” sets K in Rm and Cm and show how the smoothness of K effects the estimates.

On the asymptotic distribution of oscillation points in rational approximation

AbstractОб асимптотическом р аспределении точек о сцилляции при рациональной апп роксимации А. Кроо и Ф. Гшхерсторфё р В статье исследуется асимптотическое рас пределение точек осцилляции (аль тернанса) уклонения функции от рациональной дроби е ё наилучщего приближе ния, где степень числи теля дроби ≦т, степень знам енателя ≦(m) и л(m)<m, п(m)≦п(т+1)≦п (т) + 1. Установлено, что для каждой непрер ывной функции f сущест вует подпоследовательно сть натуральных чисел ти такая, что для нее по крайней мере т-п (т) точек из общего числа т+п(т)+2-d(m) точек осцилляции укл онения f-rm, n(m)*) асимптотич ески равномерно распреде лены относительно чебыше вской меры (здесь d(m) — по рядок вырождения дроби rm, n(m)*))/П оказано также, что этот резуль тат неулучшаем.