Balayage of Fourier Transforms and the Theory of Frames

Abstract

Every separable Hilbert space has an orthogonal basis. This allows every element in the Hilbert space to be expressed as an infinite linear combination of the basis elements. The structure of a basis can be too rigid in some situations. Frames gives us greater flexibility than bases. A frame in Hilbert space is a spanning set with the reconstruction property.

A frame must satisfy both an upper frame bound and a lower frame bound. The requirement of an upper bound is rather modest. Most of the mathematical difficulty lies in showing the lower bound exists.

We examine the theory of Beurling on Balayage of Fourier transforms and the role of spectral synthesis in this theory. Beurling showed that if the condition of Balayage holds, then the lower frame bound for a Fourier frame exists under suitable hypothesis. We extend this theory to obtain lower bound inequalities for other types of frames. We prove that lower bounds exist for generalized Fourier frames and two types of semi-discrete Gabor frames.