Phases for dyadic orthonormal wavelets
Qing Gu, Deguang Han · Journal of Mathematical Physics · 2002
We consider real-valued functions α(s) and wavelets ψ∈L2(R) such that eiα(s)|ψ̂(s)| is the Fourier transform of a wavelet. Such a function α(s) is called an attainable phase for the wavelet ψ. It is known that for all multiresolution analysis (MRA) wavelets, the phase function α(s)=12s is attainable, and any real function α(s) is attainable by any minimally-supported-frequency (MSF) wavelet. Besides this, very little is known in the literature about attainable phases for wavelets. We study the problem of determining functions which are attainable phases for some (non-MSF) wavelets. We prove that there exists a non-MSF wavelet for which there is no attainable “set-wise” linear phase. This answers a basic question about wavelet phases. Although we do not know whether for any irrational number a, as is attainable by some non-MSF wavelets, we show that there exist certain rational numbers a such that as is not attainable by any non-MSF wavelet. We also prove that there exists a large class of rational numbers a such that as is attainable by some non-MSF wavelets. We examine the relationship between different classes of wavelets admitting linear phases. In particular we present an example of a non-MSF wavelet which is not an MRA wavelet but admits linear phase 12s.