Random Sampling of Mellin Band-Limited Signals

Abstract

In this paper, we address the random sampling problem for the class of functions in the space of Mellin band-limited functions ${\mathcal{B}}_T$, which are concentrated on a bounded cube. It is established that any Mellin band-limited function can be approximated by an element in a finite-dimensional subspace of ${\mathcal{B}}_T$. Utilizing the notion of covering number and Bernstein’s inequality to the sum of independent random variables, we prove that the probabilistic sampling inequality holds for the set of concentrated signals in ${\mathcal{B}}_T$ with an overwhelming probability provided the sampling size is large enough.

Keywords:

• Mellin band-limited functions
• Mellin transform
• random sampling
• reproducing kernel space
• sampling inequality

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 94A20
• 41A05
• 42A61
• 42C15

Additional information

Funding

S. Bajpeyi and S. Sivananthan gratefully thank financial assistance from the Department of Science and Technology, Government of India, via project no. CRG/2019/002412.