Approximation and Orthogonality in Sobolev Spaces

University Of Oregon Eugene, Eugene OR

Investigators

Abstract

This research project is concerned with the study of approximations of functions of several variables by families of simpler, polynomial functions (and related questions) in the setting of Sobolev spaces, which are abstract mathematical function spaces originally developed to study problems in mathematical physics and which are utilized today in a number of scientific computing settings. A truly complex system or problem is often intractable, and we often need to find an approximation that is more manageable. Approximation methods on domains in higher dimensional spaces are good examples of this principle, and they are crucial in many problems in applied mathematics. In contrast to one dimension, many challenging problems in higher dimensions that are of fundamental importance are not resolved. The Principal Investigator will study several problems on approximation and orthogonality in Sobolev spaces that rely on new connections and ideas revealed only recently. The project aims at both theoretical understanding and construction of new approximation methods, and the work has the potential to impact scientific computing, numerical analysis, statistics, and geoscience. The Principal Investigator will study approximation and orthogonality in Sobolev spaces on regular domains, such as cubes, balls, spheres, and simplexes. The project combines several research topics: approximation theory, Fourier analysis, numerical analysis, and orthogonal polynomials. One of the main problems originates from the area of spectral methods for the numerical solution of partial differential equations. Through recent work of the Principal Investigator and collaborators, it has become increasingly clear that understanding orthogonality in Sobolev spaces is crucial for approximation and computation in Sobolev spaces. This research will be based on recent progress in characterization of best approximation by polynomials on the unit sphere and on the unit ball, in Sobolev orthogonal polynomials, and in spectral approximation. The project is expected to lead to new scientific computational methods and new algorithms.