Title

Irregular sampling with derivatives

Date of Completion

January 1995

Keywords

Mathematics

Degree

Ph.D.

Abstract

This thesis is concerned with the problem of irregular sampling with derivatives. In one dimension, we develop two types of reconstruction algorithms that are reliable, efficient, and of practical use for $f\in B\sb\Omega,$ the class of finite energy band-limited functions (signals) with bandwidth $2\Omega,$ given the samples $\{f\sp{(k)}(x\sb{n});0\le k\le p\},$ for a general $\{x\sb{n}\}.$ The algorithms of the first type follow from a well known operator approximation technique. They converge geometrically under a new sufficient condition on the maximum gap. This condition is an increasing function of p and allows gaps larger than Nyquist.^ For the second type, we produce (weighted) frames for $B\sb\Omega,$ then use the frame algorithm to deduce reconstruction schemes that converge geometrically as well. Furthermore, the weights compensate with the irregularity of the sampling set producing very stable and flexible algorithms under the same maximum gap condition as for the first type.^ For the case p = 1, we develop two finite dimensional models for numerical implementation. One is on the space of trigonometric polynomials in which case the problem can be interpreted as an efficient method for Hermite interpolation. The other is on the space of band-limited discrete periodic signals. Here, the sampling points are "pairs" so that derivative samples can be approximated by differences. This second model is a new method for discrete irregular sampling which is efficient even in the presence of gaps larger than the Nyquist gap, unlike previously known methods. We support this claim by implementing the algorithms and running some comparison experiments with two other methods.^ In several dimensions, we give explicit necessary density conditions on the sampling set to ensure a stable reconstruction from function data and partial derivatives data, the sampling set being any uniformly discrete subset of $\IR\sp{d}.$ Here, we use the definition of density introduced by A. Beurling. Furthermore, for the case d = 1 and any k, we use the supremum norm (instead of the $L\sp2$-norm) to derive a better sufficient density condition. These density conditions "generalize" the one found in uniform derivatives sampling. ^