CAREER: Bases in Hilbert Function Spaces and Some of their Applications

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This project discusses different ways by which a function can be decomposed, or approximated, by sums of functions with a simple structure. Of particular interest are decompositions into sums of the simple sine and cosine functions. For functions defined over an interval, or in higher dimensions over a cube, such decompositions are a classical area of study. This project considers functions defined over more complicated sets, and studies the existence of similar decompositions in such settings. The ability to approximate or decompose functions in such a way provides a strong tool for research, which can be applied in various settings. Several applications of this theory will be studied throughout this project. As part of this project, the PI will develop, organize and teach in a new program titled "Honors level Program in Analysis for Students in the Atlanta Area". This three-year program, for promising undergraduate students, is expected to increase both the motivation and the potential of its participants to be admitted to leading graduate programs. It is well known that most measurable sets do not admit an orthogonal basis of exponentials, or of trigonometric functions. Therefore, to use harmonic analysis machinery over such sets, the rigid structure of an orthonormal basis needs to be replaced by the more flexible structure of a Riesz basis, or if this is impossible, by relaxed versions of it such as frames and Riesz sequences. In this project, the PI will continue her study (with collaborators) of such problems. For example, she will analyze questions regarding the existence of Riesz bases of exponentials, and problems related to systems with universal properties. Further, the PI will apply this theory to mathematical problems in several different settings, including sampling and interpolation of band limited signals, extensions and refinements of certain uncertainty principles, and the behavior of Gaussian noise over long intervals of time.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.