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Seminario: “On sets of functions with variation diminishing property”

Mercoledì 5 Febbraio 2020, ore 11:00 - Aula 2AB45 - Mariusz Bieniek

ARGOMENTI: Seminari

Mercoledì 5 Febbraio 2020 alle ore 11:00 in Aula 2AB45, Mariusz Bieniek (Institute of Mathematics, Marie Curie Sklodowska University) terrà un seminario dal titolo “On sets of functions with variation diminishing property”.

Abstract
A finite set of functions $f_1,\dots, f_n$ defined on an interval $[a, b] \subset \mathbb{R}$ with values in $\mathbb{R}$ is said to have variation diminishing property (VDP) if the number of zeros in $(a, b)$ of any linear combination $\sum_{j=1}^{n}a_jf_j$ does not exceed the number of sign changes in the sequence $(a_1,\dots, a_n)$ of the coefficients of the combination. The best known examples are monomials $1, x, x^2,\dots, x^n$ on the positive half-axis $(0, \infty)$ (this is so called Descartes’ rule of signs) and Bernstein polynomials on the interval $(0, 1)$.
An excellent set of problems concerning Descartes’ rule of signs and a couple of its proofs can be found in Part V, Chapter 1 of the book of Pólya and Szego [1]. New simple proofs and historical remarks can be found in the papers of Wang [2] and Komornik [3].
During my talk I will present other sets with having VDP, including some special cases of special functions known as Meijer’s G–functions. An excellent presentation of general G-functions and its numerous properties can be found in the monograph of Mathai [4]. The talk will be a compilation of the results presented in my paper [5]. The most interesting issue is the comparison of the proofs of VDP for various sets of functions. If time permits, I will discuss some applications of VDP in statistics.

References
[1] Pólya, G., Szego, G. Problems and theorems in analysis. II. Theory of functions, zeros, polynomials, determinants, number theory, geometry. Springer-Verlag, Berlin, 1998.
[2] Wang, X., A Simple Proof of Descartes’s Rule of Signs. American Mathematical Monthly. 111 (2004), 525-526.
[3] Komornik, V., Another short proof of Descartes’s rule of signs. Amer. Math. Monthly 113 (2006), 829–830.
[4] Mathai, A. M., A Handbook of Generalized Special Functions for Statistical and Physical Sciences, Clarendon Press, Oxford, 1993.
5] Bieniek, M., Variation diminishing property for densities of uniform generalized order statistics, Metrika 65 (2007), 297-309.