Multi-window dilation-and-modulation frames on the half real line

TL;DR

This paper introduces a new class of dilation-and-modulation frames on the positive real line, characterizes their duals without Fourier transform methods, and reveals their redundancy properties based on the number of generators.

Contribution

It develops the $ heta_{a}$-transform for dilation-and-modulation frames in $L^{2}(R_{+})$, providing explicit dual frame expressions and analyzing redundancy based on generator count.

Findings

Frames with one generator are nonredundant.

Frames with multiple generators are redundant.

Explicit dual frames are characterized for general dilation-and-modulation frames.

Abstract

Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators, respectively. They have been extensively studied. However, dilation-and-modulation systems have not, and they cannot be derived from wavelet or Gabor systems. In this paper, we investigate a class of dilation-and-modulation systems in the causal signal space $L^{2}(\Bbb R_{+})$. $L^{2}(\Bbb R_{+})$ can be identified a subspace of $L^{2}(\Bbb R)$ consisting of all $L^{2}(\Bbb R)$-functions supported on $\Bbb R_{+}$, and is unclosed under the Fourier transform. So the Fourier transform method does not work in $L^{2}(\Bbb R_{+})$. In this paper, we introduce the notion of $\Theta_{a}$-transform in $L^{2}(\Bbb R_{+})$, using $\Theta_{a}$-transform we characterize dilation-and-modulation frames and dual frames in $L^{2}(\Bbb R_{+})$; and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for $L^{2}(\Bbb R_{+})$. Interestingly, we prove that an arbitrary frame of this form is always nonredundant whenever the number of the generators is $1$, and is always redundant whenever it is greater than $1$. Some examples are also provided to illustrate the generality of our results.