Uniqueness for the continuous wavelet transform

TL;DR

This paper proves conditions under which the continuous wavelet transform uniquely identifies signals, showing injectivity for square integrable signals and constant signals for bounded signals when the transform vanishes.

Contribution

It establishes new injectivity results for the continuous wavelet transform under weak Fourier domain conditions and characterizes signals with zero wavelet transform.

Findings

Injectivity holds for square integrable signals under weak Fourier conditions.

If the wavelet transform of a bounded signal is zero, the signal must be constant.

Provides theoretical foundations for signal uniqueness in wavelet analysis.

Abstract

Injectivity of the continuous wavelet transform acting on a square integrable signal is proved under weak conditions on the Fourier transform of the wavelet, namely that it is nonzero somewhere in almost every direction. For a bounded signal (not necessarily square integrable), we show that if the continuous wavelet transform vanishes identically, then the signal must be constant.