Joint dilation scaling sets on the reducing subspaces (Q2911045)

A wavelet set in \(\mathbb{R}^n\) associated to a matrix \(A\) (an \(A\)-wavelet set) is a subset of \(\mathbb{R}^n\) such that the inverse Fourier transform of the characteristic function of the set is a wavelet function for the dilation \(A\). Similarly, an \(A\)-scaling set gives a scaling function of an MRA associated to \(A\) by the inverse Fourier transform of its characteristic function.NEWLINENEWLINEIn the present paper, the authors consider \((A, B)\)-scaling sets (and \((A, B)\)-wavelet sets), that is sets that are simultaneously scaling sets (respectively, wavelet sets) for different \(n\times n\) expansive matrices \(A\) and \(B\).NEWLINENEWLINEFirst they study the one-dimensional case. Since it is known that there are neither \(2\)- nor \((-2)\)-scaling sets in \(\mathbb{R}\) having two intervals, the authors characterize three-interval \((-2)\)-scaling sets. Using this and a result of N. K. Shukla and G. C. S. Yadav in which a characterization of three-interval 2-scaling sets is given, the authors determine all those joint \((2,-2)\)-scaling sets of \(\mathbb{R}\). For wavelet sets, they provide an example of a \((-2)\)-wavelet set which is not a \(2\)-wavelet set. They also present a class of \(2\)-wavelets sets which fail to be \((-2)\)-wavelet sets employing generalized Journé wavelet sets.NEWLINENEWLINEIn the second part of this work, the authors study the existence of joint \((A, B)\)-scaling sets in subspaces of \(L^2(\mathbb{R}^n)\) of the form \(L^2_E(\mathbb{R}^n)=\{f\in L^2(\mathbb{R}^n): \operatorname{supp}(\widehat{f})\subseteq E\}\), where \(E\) is a measurable set in \(\mathbb{R}^n\) and \(A\) and \(B\) are \(n\times n\) expansive matrices with integer entries which provide \(A\)-multiwavelet sets and \(B\)-multiwavelet sets of order \(|\det(A)|-1\) and \(|\det(B)|-1\), respectively. They show how to construct joint scaling sets in higher dimensions using \((2,-2)\)-scaling sets in \(\mathbb{R}\). Their result provides a method to obtain MRA multiwavelet sets in \(\mathbb{R}^n\). Finally, the authors characterize joint \((2,-2)\)-wavelet sets in \(L^2_E(\mathbb{R})\) and give several examples that support their previous results.