Frames by Multiplication

TL;DR

This paper investigates how multiplying functions affects their frame and Riesz basis properties, with applications to irregular translates of bandlimited functions and the construction of frames using density results.

Contribution

It provides new insights into the frame properties of multiplied functions and applies these to construct frames from irregular translates of bandlimited functions.

Findings

Established conditions for frame and Riesz basis properties after multiplication.

Applied density results to construct frames from irregular translates.

Provided methods to verify frame properties in specific function sets.

Abstract

In this note we study frame-related properties of a sequence of functions multiplied by another function. In particular we study frame and Riesz basis properties. We apply these results to sets of irregular translates of a bandlimited function $h$ in $L^2(\R^d)$. This is achieved by looking at a set of exponentials restricted to a set $E \subset \R^d$ with frequencies in a countable set $\Lambda$ and multiplying it by the Fourier transform of a fixed function $h \in L^2(E)$. Using density results due to Beurling, we prove the existence and give ways to construct frames by irregular translates.