Totally Positive Kernels, Polya Frequency Functions, and Their Transforms

Explore the mathematical concepts of totally positive kernels and Polya frequency functions in this 51-minute lecture from the Fields Institute's Focus Program on Analytic Function Spaces and their Applications. Delve into topics such as total nonnegativity, real rooted polynomials, and quasi-stable polynomials. Examine composition operators, Whitney's theorem, and the characterization of total positivity. Investigate preservers of totally nonnegative kernels, the existence of totally positive kernels, and their approximations. Learn about Hankel kernels and discuss open problems in the field. Gain insights from Dominique Guillot of the University of Delaware as he presents this comprehensive overview of advanced mathematical concepts and their applications.

Intro
Total nonnegativity
Total positivity in mathematics
Examples (cont.)
Real rooted polynomials
Quasi-stable polynomials
Composition operators
Finite dimensional case: X and Y finite
From TN to TP: Whitney's theorem.
Characterization for TP
Preservers of TN kernels
Existence of TP kernels
TP kernel approximations
Restricting to special families
Back to general TP kernels
Hankel kernels
Open problems
References