A construction of a pairwise orthogonal wavelet frames using polyphase matrix (Q2901500)

The current work extends work of \textit{G. Bhatt}, \textit{B. D. Johnson} and \textit{E. Weber} [Appl. Comput. Harmon. Anal. 23, No. 2, 215--234 (2007; Zbl 1136.42026)] and of \textit{H. O. Kim, R. Y. Kim, J. K. Lim} and \textit{Z. Shen} [J. Approximation Theory 147, No. 2, 196--204 (2007; Zbl 1116.42009)]. Let \(P\) be the \(r\times 2\) matrix with trigonometric polynomial entries \(m_{k,i}(\xi)\), \(k=1,\dots,r\), \(i=0,1\), where \(m_{k,0}(\xi)=(m_k(\xi/2)+m_k(\xi/2+1/2))/\sqrt{2}\) and \(m_{k,1}(\xi) =e^{\pi i \xi} ( m_k(\xi/2)- m_k(\xi/2+1/2))/\sqrt{2}\) such that the matrix \(M\) with \(k\)th row \([m_k(\xi)\,\, m_k(\xi+1/2)]\) satisfies \(M^\ast(\xi) M(\xi)=I_2\). The filters \(m_k\) are generated by a unitary extension principle. Let \(A\) be any \(2r\times 2r\) paraunitary matrix and let \(V_1\) and \(V_2\) denote the submatrices of first \(r\) and last \(r\) columns of \(A\), respectively, and let \(P_i\), \(i=1,2\), denote the \((2r+1)\times (r+1)\) matrix obtained from \(V_i\) by appending first row \([1,0,\dots, 0]\in \mathbb{R}^{r+1}\) and first column \([1,0,\dots ,0]^T\in \mathbb{R}^{2r+1}\). Set \(\widetilde{P}_i=P_iP\) and define a new filter pair \(\widetilde{m}_k(\xi)\) and \(\widetilde{n}_k(\xi)\) by \(\widetilde{m}_k(\xi)=([\widetilde{P}_1(2\xi)]_{k,1}+e^{-2\pi i \xi}[\widetilde{P}_1(2\xi)]_{k,2})/\sqrt{2}\) and \(\widetilde{n}\) the same way with \(\widetilde{P}_1\) replaced by \(\widetilde{P}_2\). It is proved that with these definitions, the wavelet systems \(\widetilde{\psi}_k\) and \(\widetilde{\eta}_k\) defined by \(\widehat{\widetilde\psi}(2\xi)=\widetilde{m}_k(\xi)\widehat{\phi}(\xi)\) and \(\widehat{\widetilde\eta}(2\xi)=\widetilde{n}_k(\xi)\widehat{\phi}(\xi)\), \(k=1,\dots, 2r\), constitute a pair of orthogonal Parseval wavelet frames for \(L^2(\mathbb{R})\). Here \(\phi\) is the scaling function of the original MRA.