View Proposal #474
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| ID | 474 |
|---|---|
| First Name | Aqeeb |
| Last Name | Sabree |
| Institution | University of Iowa |
| Speaker Category | graduate student |
| Title of Talk | Research Topics from Reproducing Kernel Hilbert Spaces |
| Abstract | Reproducing Kernel Hilbert Spaces (RKHS) have applications to statistics, machine learning, differential equations, and more. The goal of this presentation is to introduce the concept of a RKHS, and discuss it’s applications to many research areas. The amazing thing about this research area is that there are many research questions/topics for dissertations or undergraduate research experiences. I will give a brief history of RKHSs, highlighting where it has appeared and how it has been applied. Then I will present the theoretical foundation(s) of the subject; from here I will go into its applications. Below, you will find highlights of the theory that I will present, and some highlights of its application. You can discuss the existence of RKHSs in different ways: one, you can prove that the evaluation functional is bounded; or, two, you can prove that (given a Hilbert space) the Hilbert space has a reproducing kernel function. A nice property of the reproducing kernel is that it is unique. Thus, every RKHS has exactly one reproducing kernel; furthermore, every reproducing kernel is the reproducing kernel for a unique RKHS (Moore--Aronszajn). The process of recreating the RKHS from the kernel function is termed the it reconstruction problem, and is an interesting research area. The usefulness of the theory of RKHSs can be seen in the fact that the finite energy Fourier, Hankel, sine, and cosine transformed band-limited signals are specific realizations of the abstract reproducing kernel Hilbert space (RKHS). Sampling Theory: Sampling theory deals with the reconstruction of functions (or signals) from their values (or samples) on an appropriate set of points. When given a reproducing kernel Hilbert space, H; one asks: What are some (suitable) sets of points which reproduce (or interpolate) the full values of functions from H? And when given points in a set S, one asks: What are the RKHSs for which S is a complete set of sample points? Meaning the values of functions from H are reproduced by interpolation from S. |
| Subject area(s) | Advanced Calculus would be helpful |
| Suitable for undergraduates? | Y |
| Day Preference | Either |
| Computer Needed? | Y |
| Bringing a laptop? | N |
| Overhead Needed? | N |
| Software requests | None |
| Special Needs | I will need a board to write on, and hopefully still have access to the presentation projection. |
| Date Submitted | 10/05/2017 |
| Year | 2017 |