Page 135 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 135
APPROXIMATION THEORY AND APPLICATIONS (MS-78)
Sampling strategies for approximation in kernel spaces
Gabriele Santin, gsantin@fbk.eu
Bruno Kessler Foundation, Italy
Kernel methods provide powerful and flexible techniques to approximate functions defined on
general domains, with possible high-dimensional input and output dimension, and using sam-
ples at scattered locations.
In this context, the problem of choosing the location of the sampling points is of great
interest, both from a practical and a theoretical viewpoint. On one hand, it is of theoretical
interest to know the limit and benefits of the choice of optimal point location, and to design
feasible algorithms to select them. On the other hand, several applications are described by
large datasets, and it may be interesting to select a possibly small portion of the data that allows
an accurate reconstruction of the full problem.
In this talk we will discuss some greedy methods and show that they are effective techniques
in both scenarios.
In particular, we will first introduce some results on the general structure and theory of
kernel-based greedy methods, and describe their efficient implementation. We will then show
that, in certain circumstances, they may be proven to be worst-case optimal.
We will focus mainly on interpolation, and mention some application to quadrature. More-
over, we will discuss the use of these techniques on some real world applications.
Inequalities for Legendre polynomials and applications in information
potential
Daniel Florin Sofonea, florin.sofonea@ulbsibiu.ro
"Lucian Blaga" of Sibiu, Romania
Coauthor: Ioan Tincu
The classical orthogonal polynomials play an important role in applications of mathematical
analysis, spectral method with applications in fluid dynamics and other areas of interest. This
work is devoted to the orthogonal polynomials. Bounds of Legendre polynomials are obtained
in terms of inequalities. Also, entropies associated with discrete probability distributions is a
topic considered. Bounds of the entropies which improve some previously known results are
obtained.
Gauss-Lucas theorem in polynomial dynamics
Margaret Stawiska Friedland, stawiska@umich.edu
American Mathematical Society/MathSciNet, United States
Using versions of the Gauss-Lucas theorem adapted to dynamics, we prove that for every
complex polynomial p of degree d ≥ 2 the convex hull Hp of the Julia set Jp of p satisfies
p−1(Hp) ⊂ Hp. This settles positively a conjecture by P. Alexandersson. We also characterize
the families of polynomials for which the equality p−1(Hp) = Hp is achieved.
133
Sampling strategies for approximation in kernel spaces
Gabriele Santin, gsantin@fbk.eu
Bruno Kessler Foundation, Italy
Kernel methods provide powerful and flexible techniques to approximate functions defined on
general domains, with possible high-dimensional input and output dimension, and using sam-
ples at scattered locations.
In this context, the problem of choosing the location of the sampling points is of great
interest, both from a practical and a theoretical viewpoint. On one hand, it is of theoretical
interest to know the limit and benefits of the choice of optimal point location, and to design
feasible algorithms to select them. On the other hand, several applications are described by
large datasets, and it may be interesting to select a possibly small portion of the data that allows
an accurate reconstruction of the full problem.
In this talk we will discuss some greedy methods and show that they are effective techniques
in both scenarios.
In particular, we will first introduce some results on the general structure and theory of
kernel-based greedy methods, and describe their efficient implementation. We will then show
that, in certain circumstances, they may be proven to be worst-case optimal.
We will focus mainly on interpolation, and mention some application to quadrature. More-
over, we will discuss the use of these techniques on some real world applications.
Inequalities for Legendre polynomials and applications in information
potential
Daniel Florin Sofonea, florin.sofonea@ulbsibiu.ro
"Lucian Blaga" of Sibiu, Romania
Coauthor: Ioan Tincu
The classical orthogonal polynomials play an important role in applications of mathematical
analysis, spectral method with applications in fluid dynamics and other areas of interest. This
work is devoted to the orthogonal polynomials. Bounds of Legendre polynomials are obtained
in terms of inequalities. Also, entropies associated with discrete probability distributions is a
topic considered. Bounds of the entropies which improve some previously known results are
obtained.
Gauss-Lucas theorem in polynomial dynamics
Margaret Stawiska Friedland, stawiska@umich.edu
American Mathematical Society/MathSciNet, United States
Using versions of the Gauss-Lucas theorem adapted to dynamics, we prove that for every
complex polynomial p of degree d ≥ 2 the convex hull Hp of the Julia set Jp of p satisfies
p−1(Hp) ⊂ Hp. This settles positively a conjecture by P. Alexandersson. We also characterize
the families of polynomials for which the equality p−1(Hp) = Hp is achieved.
133
