Phases for dyadic orthonormal wavelets
Abbreviated Journal Title
J. Math. Phys.
We consider real-valued functions alpha(s) and wavelets psiis an element ofL(2)(R) such that e(ialpha(s))\psi(s)\ is the Fourier transform of a wavelet. Such a function alpha(s) is called an attainable phase for the wavelet psi. It is known that for all multiresolution analysis (MRA) wavelets, the phase function alpha(s)=shape=1/2 s is attainable, and any real function alpha(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 1/2 s. (C) 2002 American Institute of Physics.
Journal of Mathematical Physics
"Phases for dyadic orthonormal wavelets" (2002). Faculty Bibliography 2000s. 3236.