A Construction of Multiwavelet Sets in the Euclidean Plane

Abstract

For \(A=\left( \begin{array}[pos]{ clrr} 0 & 1 \\ a & 0 \end{array} \right),\) where \(a\) is an integer such that \(|a|>1\) and a natural number \(d\) satisfying \(L=(|a|-1)d,\) we obtain that the product \(W\times Q\) of a measurable set \(W\) of the Lebesgue measure \(2\pi L,\) and a measurable set \(Q\) in \(\mathbb R\) such that \(Q\subset aQ,\) is an MRA \(A\)-multiwavelet set of order \(Ld\) in \(\mathbb R^2\) if and only if \(W\) is an \(a\)-multiwavelet set of order \(L\) and \(Q\) is an \(a\)-multiscaling set of order \(d\) associated with \(W.\)