The Role of Gaussian Curvature in Harmonic Analysis and Related Areas

University Of Missouri-Columbia, Columbia MO

Investigators

Abstract

ABSTRACT: The author proposes to study a set of problems in harmonic analysis and related areas where the Gaussian curvature, the determinant of the differential of the Gauss map taking the point on a hypersurface to the unit normal at that point, plays an important role. More specifically, the author proposes to study the following three basic problems: regularity of averages over surfaces, distribution of lattice points in convex domains, and the existence of orthogonal exponential bases for domains in Euclidean space. In the context of averages over surfaces, we propose to find a set of necessary and sufficient conditions for the Lebesgue space boundedness of these averages in terms of natural and easily computable geometric criteria. In the context of lattice points in convex domains, we propose to compute a sharp rate of growth for the discrepancy between the volume of a dilated convex body and the number of lattice points trapped inside, again in terms of natural geometric properties of the boundary. In the context of orthogonal exponential bases, we propose to make progress towards the proof of the Fuglede Conjecture, which says that a domain has orthogonal exponential basis if and only if it is possible to tile Euclidean space with disjoint translates of this domain. Combinatorial and number theoretic methods are expected to play an important role. The study of the maximal averaging operators and other similar operators in harmonic analysis is partially motivated by the following interesting question: How close can we come to recovering a set of data from the various kinds of averages of that data? The question is of potential practical value since scientists are often called upon to make predictions based on average information. For example, meteorologists make predictions about the rainfall in a particular location based on the average rainfall in years past in nearby towns. Seismologists make earthquake predictions based on the pattern of shocks in the surrounding area. The tradeoff involved in the study of these phenomena is, roughly speaking, the following. If the data is very precise, then it can, generally speaking, be recovered from any kind of a reasonable average. If the data is less precise, then we have to make sure that the averaging process compensates for the deficiencies of the data. The main thrust of this project is to study the averaging phenomenon when the data is given by a certain kind of a mathematical function, and the average is taken over a curved surface. The study of the distribution of lattice points in convex domains and the associated discrepancy function is motivated by the desire to approximate discrete information, for example integer points in the plane, by more easily computable continuous information, in this case the area. Finally, the study of orthogonal exponential bases is motivated by an important practical problem of approximating functions by trigonometric functions. These types of approximations have numerous applications in physics, engineering, and many other areas of science and technology.