Page 128 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 128
APPROXIMATION THEORY AND APPLICATIONS (MS-78)
A generalization of a local form of the classical Markov inequality
Tomasz Beberok, t_beberok@pwsztar.edu.pl
University of Applied Sciences in Tarnow, Poland
In this talk we introduce a generalization to compact subsets of certain algebraic varieties of the
classical Markov inequality on the derivatives of a polynomial in terms of its own values. We
also introduce an extension to such sets of a local form of the classical Markov inequality, and
show the equivalence of introduced Markov and local Markov inequalities.
Metric Fourier approximation of set-valued functions of bounded
variation
Elena Berdysheva, elena.berdysheva@math.uni-giessen.de
Justus Liebig University Giessen, Germany
Coauthors: Nira Dyn, Elza Farkhi, Alona Mokhov
We study set-valued functions (SVFs) mapping a real interval to compact sets in Rd. Older ap-
proaches to the approximation investigated almost exclusively SVFs with convex images (val-
ues), the standard methods suffer from convexification. In this talk I will describe a new con-
struction that adopts the trigonometric Fourier series to set-valued functions with general (not
necessarily convex) compact images. Our main result is analogous to the classical Dirichlet-
Jordan Theorem for real functions. It states the pointwise convergence in the Hausdorff metric
of the metric Fourier partial sums of a multifunction of bounded variation to a set determined
by the values of the metric selections of the function. In particular, if the multifunction F is of
bounded variation and continuous at a point x, then the metric Fourier partial sums of it at x
converge to F (x).
Piecewise-regular approximation of maps into real algebraic sets
Marcin Bilski, marcin.bilski@im.uj.edu.pl
Jagiellonian University, Poland
Coauthor: Wojciech Kucharz
A real algebraic set W of dimension m is said to be uniformly rational if each of its points has
a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of Rm.
Let l be any nonnegative integer. It turns out that every map of class Cl from a compact subset
of a real algebraic set into a uniformly rational real algebraic set can be approximated in the Cl
topology by piecewise-regular maps of class Ck, where k is an arbitrary integer greater than or
equal to l.
126
A generalization of a local form of the classical Markov inequality
Tomasz Beberok, t_beberok@pwsztar.edu.pl
University of Applied Sciences in Tarnow, Poland
In this talk we introduce a generalization to compact subsets of certain algebraic varieties of the
classical Markov inequality on the derivatives of a polynomial in terms of its own values. We
also introduce an extension to such sets of a local form of the classical Markov inequality, and
show the equivalence of introduced Markov and local Markov inequalities.
Metric Fourier approximation of set-valued functions of bounded
variation
Elena Berdysheva, elena.berdysheva@math.uni-giessen.de
Justus Liebig University Giessen, Germany
Coauthors: Nira Dyn, Elza Farkhi, Alona Mokhov
We study set-valued functions (SVFs) mapping a real interval to compact sets in Rd. Older ap-
proaches to the approximation investigated almost exclusively SVFs with convex images (val-
ues), the standard methods suffer from convexification. In this talk I will describe a new con-
struction that adopts the trigonometric Fourier series to set-valued functions with general (not
necessarily convex) compact images. Our main result is analogous to the classical Dirichlet-
Jordan Theorem for real functions. It states the pointwise convergence in the Hausdorff metric
of the metric Fourier partial sums of a multifunction of bounded variation to a set determined
by the values of the metric selections of the function. In particular, if the multifunction F is of
bounded variation and continuous at a point x, then the metric Fourier partial sums of it at x
converge to F (x).
Piecewise-regular approximation of maps into real algebraic sets
Marcin Bilski, marcin.bilski@im.uj.edu.pl
Jagiellonian University, Poland
Coauthor: Wojciech Kucharz
A real algebraic set W of dimension m is said to be uniformly rational if each of its points has
a Zariski open neighborhood which is biregularly isomorphic to a Zariski open subset of Rm.
Let l be any nonnegative integer. It turns out that every map of class Cl from a compact subset
of a real algebraic set into a uniformly rational real algebraic set can be approximated in the Cl
topology by piecewise-regular maps of class Ck, where k is an arbitrary integer greater than or
equal to l.
126
