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Leokadia Bialas-Ciez

Publications and source records attributed to Leokadia Bialas-Ciez.

6 recordsLinked to original sources
arXiv2025

Polynomial meshes on algebraic sets

Polynomial meshes (called sometimes "norming sets") allow us to estimate the supremum norm of polynomials on a fixed compact set by the norm on its discrete subset. We give a general construction of polynomial weakly admissible meshes on compact subsets of arbitrary algebraic hypersurfaces in C^{N+1}. They are preimages by a projection of meshes on compacts in C^N. The meshes constructed in this way are optimal in some cases. Our method can be useful also for certain algebraic sets of codimension greater than one. To illustrate applications of the obtained theorems, we first give a few examples and finally report some numerical results. In particular, we present numerical tests (implemented in Matlab), concerning the use of such optimal polynomial meshes for interpolation and least-squares approximation, as well as for the evaluation of the corresponding Lebesgue constants.

math.NA↗
arXiv2023

Łojasiewicz exponent and pluricomplex Green function on algebraic sets

We study pluricomplex Green functions on algebraic sets. Let $f$ be a proper holomorphic mapping between two algebraic sets. Given a compact set $K$ in the range of $f$, we show how to estimate the pluricomplex Green functions of $K$ and of $f^{-1}(K)$ in terms of each other, the Łojasiewicz exponent of $f$ and the growth exponent of $f$. This result leads to explicit examples of pluricomplex Green functions on algebraic sets. We also present an enhanced version of the Bernstein-Walsh polynomial inequality specific to algebraic sets. This article provides a theoretical framework for future investigations of the rate of polynomial approximation of holomorphic functions on algebraic sets in the style of Bernstein-Walsh-Siciak theorem.

math.CV↗
arXiv2018

On Lagrange polynomials and the rate of approximation of planar sets by polynomial Julia sets

We revisit the approximation of nonempty compact planar sets by filled-in Julia sets of polynomials developed by Lindsey and Younsi and analyze the rate of approximation. We use slightly modified fundamental Lagrange interpolation polynomials and show that taking certain classes of nodes with subexponential growth of Lebesgue constants improves the approximation rate. To this end we investigate properties of some arrays of points in $\mathbb{C}$. In particular we prove subexponential growth of Lebesgue constants for pseudo Leja sequences with bounded Edrei growth on finite unions of quasiconformal arcs. Finally, for some classes of sets we estimate more precisely the rate of approximation by filled-in Julia sets in Hausdorff and Klimek metrics.

math.CV↗
arXiv2013

Jackson's inequality in the complex plane and the Lojasiewicz-Siciak inequality of Green's function

We prove a generalization of Dunham Jackson's famous approximation inequality to the case of compact sets in the complex plane admitting both upper and lower bounds for their Green's functions, i.e. the well known Holder Continuity Property (HCP) and the less known but crucial Lojasiewicz-Siciak inequality (LS). Moreover, we show that (LS) is a necessary condition for our Jackson type inequality.

math.CV↗
arXiv2013

Hölder continuity of pluricomplex Green function and Markov brothers' inequality

Let V_E be the pluricomplex Green function associated to a compact subset E of C^N. The well known Hölder Continuity Property of E means that there exist constants B > 0, 0< c =< 1 such that V_E(z) =< B dist(z,E)^c. The main result of this paper says that this condition is equivalent to a Vladimir Markov type inequality, i.e. || D^αP ||_E =< M^{|α|} (deg P)^{m|α|} (|α|!)^{1-m} ||P||_E, where m,M>0 are independent of the polynomial P of N variables. We give some applications of this equivalence and we present its generalization related to a notion of a fit majorant. Moreover, as a consequence of the main result we obtain a criterion for the Hölder Continuity Property in several complex variables of the type of Siciak's L-regularity criterion.

math.CV↗