Approximation, Equilibrium Measures and Discrepancy over Domains, Finite Fields and Smooth Manifolds

Georgia Southern University Research And Service Foundation, Inc, Statesboro GA

Investigators

Abstract

ABSTRACT The research will primarily focus on three areas: discrepancy estimates over domains and manifolds for classes of smooth and singular functions and kernels which depend only on distances between points in Euclidean space; the study of low discrepancy sequences such as configurations interacting via a pairwise repulsive interaction on a fixed manifold and bases of linear independent vectors over a finite field; the study of tools for good approximation of projective, group invariant and singular operators on domains and smooth manifolds. The research is expected to develop integration estimates over compact homogenous manifolds using energy functions and zonal kernels; produce theorems on separation and mesh norm properties of Riesz configurations as well as exact formulas for maximal independent sets of vectors over finite fields of fixed order; produce theorems on hyperinterpolation operators and supports of equilibrium measures. In most phases of the research, approximation theory and potential theory are expected to be useful tools. The research will primarily focus on three areas: (i) the approximation of multidimensional integrals and the study of functions which describe similarities in a finite set of data; (ii)the study of minimal energy (ground state) particles interacting via a repulsive force and linear codes with certain parameters; (iii) the study of a broad range of tools which arise in approximation theory and orthogonal polynomials. Regarding (i), our goal is to develop approximation estimates of multidimensional integrals using distance only depending functions. The former arise in the analysis of satellite data on the surface of the earth and in mathematical finance while the later occur in meaningful structures and descriptions of large data sets for example in imaging and wireless networks. Regarding (ii), our goal is to study clustering properties of minimal energy particles which are useful in understanding best packing and in the understanding of the physics of self-assembling materials. We also expect to prove the existence of linear codes which areuseful for the problem of transmitting information effectively and accurately. Regarding (iii), our goal is to understand interpolation operators on manifolds as a means to solving differential equations and supports of minimizers on the circle which are used in random matrix theory in mathematical physics.