On Supersets of Wavelet Sets |
| |
Authors: | C Viriyapong S Sumetkijakan |
| |
Institution: | 1. Department of Mathematics, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand
|
| |
Abstract: | Considering a single dyadic orthonormal wavelet ψ in L 2(?), it is still an open problem whether the support of $\widehat{\psi}$ always contains a wavelet set. As far as we know, the only result in this direction is that if the Fourier support of a wavelet function is “small” then it is either a wavelet set or a union of two wavelet sets. Without assuming that a set S is the Fourier support of a wavelet, we obtain some necessary conditions and some sufficient conditions for a “small” set S to contain a wavelet set. The main results, which are in terms of the relationship between two explicitly constructed subsets A and B of S and two subsets T 2 and D 2 of S intersecting itself exactly twice translationally and dilationally respectively, are (1) if $A\cup B\not\subseteq T_{2}\cap D_{2}$ then S does not contain a wavelet set; and (2) if A∪B?T 2∩D 2 then every wavelet subset of S must be in S?(A∪B) and if S?(A∪B) satisfies a “weak” condition then there exists a wavelet subset of S?(A∪B). In particular, if the set S?(A∪B) is of the right size then it must be a wavelet set. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|