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.