An Overview of Reproducing Kernel Hilbert Spaces

Abstract: A reproducing kernel Hilbert space is a Hilbert space of functions in which point evaluation is a continuous linear functional.  Every such space has a collection of functions called reproducing kernels.  Reproducing kernels first appeared in the work of S. Zaremba on boundary value problems for harmonic and biharmonic functions.  Later the theory was expanded by Aronszajn, Bergman, and Schwartz.  Despite origins in harmonic function theory, reproducing kernels have found applications in a wide variety of areas, including complex analysis, function approximation, machine learning, and differential equations.   In this talk, we will give an overview of some of the important theories in reproducing kernel Hilbert spaces and highlight some applications. 

Speaker’s Bio: Benjamin Russo received his Ph.D. in Mathematics from the University of Florida in operator theory and functional analysis.  He is currently a post-doc in the Data Analysis and Machine Learning group.  His current research efforts are directed towards a blend of both applied and pure functional analysis, kernel methods, and matrix theory.  At the Oak Ridge National Laboratory, he has been interested in the development of novel and robust functional analytic approaches to learning theory in dynamical systems with applications to streaming compression of scientific data.