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.
Komornik, V., Another short proof of Descartes’s rule of signs. Amer. Math. Monthly 113 (2006), 829–830.
3. Mathai, A. M., A Handbook of Generalized Special Functions for Statistical and Physical Sciences, Clarendon Press, Oxford, 1993.
4. Bieniek, M., Variation diminishing property for densities of uniform generalized order statistics, Metrika 65 (2007), 297-309.