Bounded Mean Oscillation and Bandlimited Interpolation in the Presence of Noise

TL;DR

This paper investigates how bounded, random noise affects bandlimited interpolation, showing that under certain conditions, the interpolated function maintains bounded mean oscillation and almost surely has a bounded global average.

Contribution

It introduces a generalized WSK series for bounded sequences and analyzes the properties of bandlimited functions with bounded mean oscillation in noisy settings.

Findings

Interpolated functions with random noise have bounded global averages almost surely.

Bandlimited functions with bounded mean oscillation are either bounded or have unbounded samples.

The paper provides examples illustrating the properties of such functions.

Abstract

We study some problems related to the effect of bounded, additive sample noise in the bandlimited interpolation given by the Whittaker-Shannon-Kotelnikov (WSK) sampling formula. We establish a generalized form of the WSK series that allows us to consider the bandlimited interpolation of any bounded sequence at the zeros of a sine-type function. The main result of the paper is that if the samples in this series consist of independent, uniformly distributed random variables, then the resulting bandlimited interpolation almost surely has a bounded global average. In this context, we also explore the related notion of a bandlimited function with bounded mean oscillation. We establish some properties of such functions, and in particular, we show that they are either bounded or have unbounded samples at any positive sampling rate. We also discuss a few concrete examples of functions that demonstrate these properties.