Ring-like structures of frequency domains of wavelets

It is well known that the global frequency domain Ω of any orthonormal wavelet has a hole which contains the origin, viz. the frequency domain Ω possesses a ring-like structure . We show that under some weak conditions, the set S and the hole S∗ are determined uniquely by Ω, where the size of the hole S∗ satisfies and the union of 4πν-translations (ν∈Zd) of S is the whole space Rd. 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.