Page 127 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 127
APPROXIMATION THEORY AND APPLICATIONS (MS-78)
1 ∞ − + . . . +V1 V2 (−1)p+1 Vp
|V | rps
constant of this equation is r1s r2s , where V = V (r1, r2, . . . , rp)
s=1
and Vk = V (r1, . . . , rk−1, rk+1, . . . , rp), 1 ≤ k ≤ p, are Vandermonde determinants.
A generalization of extremal functions and polynomial inequalities
Mirosław Baran, miroslaw.baran.tarnow@gmail.com
Pedagogical University of Cracow, Poland
Consider a normed space (P(CN ), N ) of polynomials of N variables equipped with a fixed
norm N , which can be arbitrary. We can define a radial version of a polynomial extremal
function, which has a sense in a general situation. In the case of the supremum norm, N (P ) =
sup{|P (z)| : z ∈ E} our extremal functions are a radial modification of the classical Siciak’s
extremal function Φ(E, z). In this special case we can also consider a local radialization of
the Siciak’s extremal function and its logarithm V (E, z) (the pluricomplex Green function).
We shall show connections between the behaviour of such extremal functions and polynomial
inequalities of Markov’s and Bernstein’s type. In particular, there will obtained new results on
Bernstein’ inequality involving higher derivatives of polynomials at interior points of compact
subsets of RN .
A complex funtion theory for Mellin Analysis and applications to sampling
Carlo Bardaro, carlo.bardaro@unipg.it
University of Perugia, Italy
Coauthors: Paul Butzer, Ilaria Mantellini, Gerhard Schmeisser
The aim of this research is to extend in a simple way the well-known Paley–Wiener theorem
of Fourier Analysis, which characterizes the so-called bandlimited functions, to the setting of
Mellin transform. In order to do that, we introduced in these last years a notion of polar analytic
function [1], which provides a simple way of describing functions that are analytic on a part
of the Riemann surface of the logarithm. Applications to various topics of Mellin Analysis are
developed, in particular sampling type theorems and quadrature formulae (for a survey see [2]).
References
[1] Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Development of a new concept
of polar analytic functions useful in Mellin analysis. Complex Var. Elliptic Equ. 64(12),
2040–2062 (2019)
[2] Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Polar-analytic functions: old and
new results, applications. Pre-print submitted, (2021)
125
1 ∞ − + . . . +V1 V2 (−1)p+1 Vp
|V | rps
constant of this equation is r1s r2s , where V = V (r1, r2, . . . , rp)
s=1
and Vk = V (r1, . . . , rk−1, rk+1, . . . , rp), 1 ≤ k ≤ p, are Vandermonde determinants.
A generalization of extremal functions and polynomial inequalities
Mirosław Baran, miroslaw.baran.tarnow@gmail.com
Pedagogical University of Cracow, Poland
Consider a normed space (P(CN ), N ) of polynomials of N variables equipped with a fixed
norm N , which can be arbitrary. We can define a radial version of a polynomial extremal
function, which has a sense in a general situation. In the case of the supremum norm, N (P ) =
sup{|P (z)| : z ∈ E} our extremal functions are a radial modification of the classical Siciak’s
extremal function Φ(E, z). In this special case we can also consider a local radialization of
the Siciak’s extremal function and its logarithm V (E, z) (the pluricomplex Green function).
We shall show connections between the behaviour of such extremal functions and polynomial
inequalities of Markov’s and Bernstein’s type. In particular, there will obtained new results on
Bernstein’ inequality involving higher derivatives of polynomials at interior points of compact
subsets of RN .
A complex funtion theory for Mellin Analysis and applications to sampling
Carlo Bardaro, carlo.bardaro@unipg.it
University of Perugia, Italy
Coauthors: Paul Butzer, Ilaria Mantellini, Gerhard Schmeisser
The aim of this research is to extend in a simple way the well-known Paley–Wiener theorem
of Fourier Analysis, which characterizes the so-called bandlimited functions, to the setting of
Mellin transform. In order to do that, we introduced in these last years a notion of polar analytic
function [1], which provides a simple way of describing functions that are analytic on a part
of the Riemann surface of the logarithm. Applications to various topics of Mellin Analysis are
developed, in particular sampling type theorems and quadrature formulae (for a survey see [2]).
References
[1] Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Development of a new concept
of polar analytic functions useful in Mellin analysis. Complex Var. Elliptic Equ. 64(12),
2040–2062 (2019)
[2] Bardaro, C., Butzer, P.L., Mantellini, I., Schmeisser, G.: Polar-analytic functions: old and
new results, applications. Pre-print submitted, (2021)
125
