Title: ‘Gabor density bounds, woven systems and weighted oblique dual frames’

Date:  Monday, 27th April

Time: 10.30am (EST)

Zoom link: https://gatech.zoom.us/j/5430145654?pwd=E6D8k4UWjLaOURjO0RyzvsyMVFtyOx.1&omn=95297390163

Committee members:

Dr. Christopher Heil (advisor)

Dr. Michael Lacey

Dr. Doron Lubinsky

Dr. Michael Damron

Dr. Kasso Okoudjou

Abstract:

The first chapter focuses on density bounds. Beurling density bounds for basis-type systems have been studied for several decades, in the frameworks of exponential systems, Gabor systems, wavelets and more general reproducing kernel Hilbert spaces. In the work of Olevskii and Ulanovskii (2009), and Nitzan (2024), necessary density conditions were established for a new class of systems known as ‘near-uniformly minimal’ exponentials. Near-uniform minimality is a generalization of uniform minimality (which itself can be thought of as an analogue of linear independence). We deduce analogous necessary density conditions for near-uniformly minimal Gabor systems. Completeness is a dual property to minimality. Much in the same way, Nitzan (2024) introduced a dual notion of ‘uniformly complete’ exponentials. She also proved necessary density conditions in this case, using a duality argument, connecting it to near-uniform minimality. We establish analogous density bounds for uniformly complete Gabor systems. However, we use a different technique, since duality fails for Gabor systems over the real line. We extend many of our results relating to near-uniform minimality/completeness, to select reproducing kernel Hilbert spaces.

In the next chapter, we study woven weighted exponential systems. A classical problem that has been extensively studied, is the basis structure problem of weighted exponentials. We consider analogous questions in the woven setting. As the name suggests, given two weights f and g, a weaving is a sequence where some of the vectors are taken from the first weighted exponential system, and the remaining are taken from the other. In our setup the standard ordering is imposed on the integers. Casazza, Cabrelli and Molter have studied woven systems in abstract Hilbert space. Our focus is specifically on woven weighted exponentials in L^2([0,a]). We obtain a complete characterization of wovenly complete systems, based on the properties of the two weights f and g. We also find a characterization for woven Bessel systems and orthonormal bases. In the case of woven frames/Riesz bases, we obtain sufficient conditions, involving a suitable perturbation requirement on g relative to f.

In the final chapter, we study oblique dual frames and weighted frames in Hilbert space. Given two subspaces of a Hilbert space, W and V, such that W and the orthogonal complement of V form a direct sum, we can define oblique duals. Oblique duals generalize frame duals, in the sense, that for a frame {f_n} in a subspace W, an oblique dual is a frame {g_n} living in V , satisfying a decomposition condition for elements in W.  We define a more general notion known as ‘weighted oblique dual frames’, where the usual oblique dual decomposition becomes a weighted sum. When the weight sequence is {1}, one gets the usual oblique dual. Analogous to the known results for oblique duals, we obtain a complete characterization of weighted oblique duals. We also study their existence when the weight sequence may be degenerate. Another avenue we explore, includes proving weighted potential and coherence results for weighted oblique dual frames. Finally, we establish a connection between the existence of weighted oblique dual frames, and the weighted frame problem.