Doctoral students at the Faculty of Technology at Linnaeus University, but other students are also welcome.
Send e-mail message by Feb 5 to ali.sirma.extern@lnu.se
1MA501 Probability Theory and Statistics 7,5 credits or an equivalent course in mathematics, mathematical statistics, or statistics.
Teaching consists of lectures, presentations and tutoring.
In this course, we will begin with a careful and rigorous introduction to RKHS, starting with the first principles and classical examples. This course is also designed as a theoretical foundation for a follow-up course that will focus on some of the applications of RKHS in for instance power series on balls and pull-backs, statistics and machine learning, integral operators, and stochastic processes.
Content
- Classical examples of RKHS: including Hardy and Bergman spaces, Paley-Wiener spaces, Sobolev spaces; illustrating how abstract theory manifests in concrete function spaces.
- Fundamental results: Hilbert space structure; characterization of reproducing kernels; reconstruction problem; the RKHS induced by a function; the RKHS of the min function; the RKHS induced by a positive matrix; the RKHS induced by an inner product.
- Interpolation and approximation: interpolation in an RKHS; strictly positive kernels; best square approximants; the elements of H(K).
After successfully completing the course, the student is anticipated to
- develop a rigorous understanding of the fundamental theory of Reproducing Kernel Hilbert Spaces, including reproducing kernels, positive definite functions, and classical examples arising in functional analysis.
- prepare students for advanced study and applications of RKHS, particularly as a theoretical foundation for kernel-based methods in machine learning and related fields.
Literature: Vern I. Paulsen, Mrinal Raghupathi, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces.
