Harmonic Analysis and Applications

June 4-8, 2018


"Stable set of sampling in a shift invariant space with multiple generators"

Kumar, Sarvesh

The aim of this paper is to find a discrete set $X = \{x_k: k \in \mathbb{Z}\}$ for a given class of functions $V$ such that any function $f$ in $V$ can be reconstructed uniquely and stable from its sample $\{f(x_k) : k \in \mathbb{Z}\}.$ The problem is motivated from the fundamental interest of digital electronics where one would like to convert the analog signals into a digital signal and vice versa for the signal transmission or the storage. The standard choice of $V$ is shift-invariant space which plays an important role in approximation theory, time-frequency analysis, digital signal processing, digital image processing and so on. To the best of our knowledge, the stable set of sampling (explicitly) is not explored in $V(\Phi),$ where $\Phi = \{\phi^1, \ldots, \phi^r\}.$ In fact, it is not known that whether the sample set $X=\mathbb{Z}$ is a stable set of sampling $V(\Phi).$ As a consequence, we obtain a surprising result that the set of integers $\mathbb{Z}$ cannot be a stable set of sampling for $V(\Phi).$ Further, we obtain an interesting example based on piecewise Hermite cubics as generators $\{\phi^1, \phi^2\}$ which yields that the set $\mathbb{Z}/3$ is a stable set of sampling but the set $\mathbb{Z}/2$ is not a stable set of sampling for $V(\phi^1, \phi^2).$ In practice, the local reconstruction from a finite number of nonuniform samples is one of the most desirable problems for many applications in signal processing. In this paper, we provide an algorithm for the local reconstruction method and illustrate with an example along with implementation using Matlab. This joint work with Prof. R. Radha and Dr. S. Sampath.

« back