Ring-like structures of frequency domains of wavelets

It is well known that the global frequency domain Omega of any orthonormal wavelet has a hole which contains the origin, viz, the frequency domain 57 possesses a ring-like structure Omega = S \S* (0 is an element of S* subset of S). We show that under some weak conditions, the set S and the hole S. are determined uniquely by Omega, where the size of the hole S. satisfies 0 is an element of (S)/(4) subset of S* subset of (S)/(2) and the union of 4 pi nu-translations (nu is an element of Z(d)) of S is the whole space R(d). Meanwhile, we give the corresponding converse theorem. We also show an interesting result: there is no orthonormal wavelet whose global frequency domain is the difference set of two balls. Finally, in order to explain our theory, we construct various global frequency domains and explain a general method of the construction of a wavelet with a given frequency domain. (C) 2009 Elsevier Inc. All rights reserved.