Abstract:
This thesis consists of six papers in the field of time-frequency analysis and sampling
theory. The common thread is the so-called short-time Fourier transform (STFT), an
integral transform that arises as the representation coefficient of the Schrödinger
representation of the Heisenberg group.
We study four types of problems, namely, (i) sampling and the existence of new Gabor
frames, (ii) optimization of point configurations, (iii) phase retrieval of the short-time
Fourier transform, and (iv) uncertainty principles for joint time-frequency
representations.
We construct new Gabor frames for two types of windows: Hermite functions and
periodic exponential B-splines. For the Hermite functions, we rely on the Janssen
representation of the Gabor frame operator, specifically, the Janssen test, to establish
new Gabor frames for Hermite functions. For the periodic exponential B-splines, we
derive a near-optimal result on sampling with derivatives in shift-invariant spaces
generated by such windows. As a direct consequence, we also obtain new Gabor frames
for periodic exponential B-splines and separated sets with density arbitrarily close to the
critical density.
We also investigate the frame bound behaviour of Gabor frames on rectangular lattices.
In particular, we show that if the window is the hyperbolic secant, an eigenfunction of the
Fourier transform, then among all rectangular lattices of integer density, the square
lattice is the unique optimizer of both bounds, and thus has the best condition number.
We employ similar techniques to provide new insight into the open problem of universal
optimality in two dimensions. In particular, we exclude that the honeycomb structure, the
natural non-lattice contender of the conjectured optimal hexagonal lattice, is universally
optimal.
We continue with the phase retrieval problem for the short-time Fourier transform: a
nonlinear non-stable problem that deals with the uniqueness of phaseless samples of the
short-time Fourier transform. We focus solely on the recovery of compactly supported
functions from phaseless samples of their short-time Fourier transform with respect to
particularly chosen holomorphic windows, not necessarily the Gaussian window.
We conclude with uncertainty principles for metaplectic time-frequency representations
of square-integrable functions in the spirit of Benedicks's theorem. While such an
uncertainty principle already exists for a subclass of metaplectic time-frequency
representations, includingthe classical time-frequency representations, there was a lack
of an understanding of the underlying group structure which would include all existing
results as special cases in a unified framework for metaplectic time-frequency
representations. We characterize the structure of metaplectic operators whose associated
time-frequency representation allows for a Benedicks-type of an uncertainty principle,
both in the sesquilinear and the quadratic version.