Accuracy of multivariable refinable functions and two-scale similarity transform (Q1578866)

The paper considers refinable function vectors \(\Phi\) with dilation matrix \(M=\text{diag}(2,2,\dots, 2)\). Let \(m(\xi)\) be a refinement symbol of the refinable vector \(\pmb\Phi\) and let \(\Phi\) provide accuracy \(p\). Then, under some additional conditions, the vector \(\pmb\Psi\) with refinement symbol \[ \tilde m(\xi)=T( 2\xi)^{-1}m(\xi)T(\xi) \] also provides accuracy \(p\), if \(T(\xi)\) is a \(2\pi\)-periodic, invertible matrix, i.e., the so-called two-scale similarity transform keeps the accuracy.