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Ring-like structures of frequency domains of wavelets (with Z. Zhang), *Applied and Computational Harmonic Analysis*, vol.29, no.1, pp.18-29, 2010.

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Abstract

It is well known that the global frequency domain of any orthonormal wavelet has
a hole which contains the origin, viz. the frequency domain possesses a
ring-like structure
*Ω = S \ S*_{*} (0 ∈ S_{*} ⊂ S).
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 *0 ∈ S/4 ⊂ S*_{*} ⊂ S/2
and the union of *4πν*-translations (*ν* ∈ **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.
**Keywords:** global frequency domain, orthonormal wavelet, regular set

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