Sampling in Spectral Subspaces and Variable Bandwidth

The bandwidth of a signal is its maximal frequency. When listening to a piece of music, it is obvious that the maximal frequency will depend on time. Thus, a signal of variable bandwidth makes perfect sense to a layperson. To the engineer and to the mathematician, however, variable bandwidth poses a conceptual problem, because the uncertainty principle presents a fundamental obstruction. In the literature one finds several notions of variable bandwidth with their merits and weaknesses. In this proposal, we will Investigate a completely new concept of variable bandwidth introduced by us recently. This notion is by means of the spectrum of certain differential operators. In mathematical jargon, a space of functions of variable bandwidth is defined as a spectral subspace of a second order elliptic differential operator. The objective of this proposal is the thorough mathematical investigation of this new concept and the design of algorithms that can be used in signal processing applications. The basic problems to be studied are sampling theorems, the existence of a critical density, and computational reconstruction algorithms. Sampling theory studies the question whether, when, and how a function or signal can be reconstructed completely from discrete samples. For example, in essence, music is stored on a CD by discrete samples. The reconstruction amounts to creating an analog signal from these samples. The necessary amount of information for a complete reconstruction depends on the bandwidth and is usually called the Nyquist rate (or critical density). We will study these questions of sampling and reconstruction for functions of variable bandwidth on several layers of generality and difficulty. Technically we will investigate sampling theorems for functions of variable bandwidth of one variable and then of several variables. By structural similarity, our approach lends itself to investigate band-limited signals on graphs, which has become an important topic in data science.

In this project we studied different approaches to variable bandwidth and corresponding sampling theorems. Bandwidth is the maximum frequency of a function, or in engineering language, of a signal. The main questions are how to recover a function from discrete samples and how much information is required for a complete reconstruction. For classical bandlimited functions these questions were answered by C. Shannon, and his answers formed the basis of information theory and digital/analog conversion in modern devices. Clearly, in a time-varying signal, such as music or speech, one can discern a changing maximum frequency, i.e., a variable bandwidth. Although there have been many attempts in engineering and physics to define and work with variable bandwidth, there is no consensus about the correct definition of variable bandwidth. The mathematical formulation is rather subtle, because the uncertainty principle presents a fundamental obstruction local frequency and variable bandwidth. In this project were studied several possible concepts of variable bandwidth, the corresponding sampling theorems (D/A conversion), and the necessary amount of information required for a reconstruction from discrete samples (necessary density conditions). (i) The first approach starts with a general representation of functions (signals) by infinite sums where each term possesses an approximate localization and frequency. Precisely, the main idea was to use series expansions with respect to Wilson bases, which were also used prominently in the signal processing for the detection of grativational waves. The main results show that the frequency truncation of Wilson expansion leads to a viable notion of variable bandwidth that is close to the intuition. (ii) The second approach replaces frequency (as it arises in the Fourier transform) by eigenvalues and eigenfunctions of operators that are motivated from physics and geometry, for instance eigenmodes of the stationary heat equation in nonhomogeneous media. Bandwidth arises by using modes (spectral parameters) with the lowest eigenvalues, a space of variable bandwidth is then a so-called spectral subspace of the corresponding operator. In this more abstract approach, the main results cover necessary density conditions that explain how much information is required to recover a function of variable bandwidth completely. (iii) Furthermore, the question of sampling and reconstruction was treated for signal models that are generated by spline functions or by a fundamental class of functions, called totally positive functions. In several cases were obtained the optimal results regarding the necessary information required for reconstruction. In higher dimension sampling theorems were studied for spaces of polynomials. The methods are drawn from time-frequency analysis, complex analysis, spectral theory, and the theory of partial differential equations, in particular elliptic differential equations.