Dilation-and-modulation systems on the half real line

Yun-Zhang Li,Wei Zhang

doi:10.1186/s13660-016-1124-y

Abstract

Translation, dilation, and modulation are fundamental operations in wavelet analysis. Affine frames based on translation-and-dilation operation and Gabor frames based on translation-and-modulation operation have been extensively studied and seen great achievements. But dilation-and-modulation frames have not. This paper addresses a class of dilation-and-modulation systems in $L^{2}(\mathbb {R}_{+})$ . We characterize frames, dual frames, and Parseval frames in $L^{2}(\mathbb {R}_{+})$ generated by such systems. Interestingly, it turns out that, for such systems, Parseval frames, orthonormal bases, and orthonormal systems are mutually equivalent to each other, while this is not the case for affine systems and Gabor systems.

Highlights

Before proceeding, we recall some notions and notations
An at most countable sequence {ei}i∈I in a separable Hilbert space H is called a frame for H if there exist < C ≤ C < ∞ such that
This paper focuses on the following systems that are like ( . ):

Summary

Introduction

We recall some notions and notations. An at most countable sequence {ei}i∈I in a separable Hilbert space H is called a frame for H if there exist < C ≤ C < ∞ such that. I∈I where C , C are called frame bounds; it is called a Bessel sequence in H if the right-hand side inequality in It is well known that {ei}i∈I and {ei}i∈I form a pair of dual frames for H if they are Bessel sequences and satisfy The Fourier transform of c ∈ l (Z) is defined by c(·) = m∈Z c(m)e– πim·. Wavelet frames in H (R) of the form {Daj Tmφ : j, m ∈ Z} were studied in [ , ], and some variations can be found in [ – ]. ). Section is devoted to Parseval frames and orthonormal bases for L (R+) of the form ). It turns out that Parseval frames, orthonormal bases, and orthonormal systems in L (R+) of the form

Proof Since a

Then f is well defined since a

This leads to