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Orthogonal wavelet frames and vector-valued wavelet transforms. (English) Zbl 1136.42026
A pair of Bessel sequences \(\{x_j\}\) and \(\{y_j\}\) in a separable Hilbert space \(H\) is said to be orthogonal if the composition operator \( \sum_j \langle \cdot,\;x_j\rangle \;y_j=0\). The direct sum sequence \(\{x_j^1\oplus\cdots\oplus x_j^N\}\) is a Parseval frame for the \(N\)-fold direct sum \(H\oplus H\oplus\dots H\) if and only if \(\{x_j^k\}_j\) is a Parseval frame for \(H\) (meaning \(\sum_j \langle \cdot,\;x_j^k \rangle \;x_j = I\) for each \(k=1,\dots,N\)), and the sequences \(\{x_j^k\}_j\) are orthogonal for different \(k\).
Given functions \(m_0,\dots, m_r\in L^\infty [0,1)\) let \(M(\xi)\) denote the \(r\times 2\) matrix whose first column is the transpose of \((m_0(\xi),\dots,m_r(\xi))\) and whose second column is that of \((m_0(\xi+1/2),\dots, m_r(\xi+1/2))\) and let \(\widetilde{M}\) be the submatrix obtained by deleting the first row of \(M\). A refinable function (one whose Fourier transform satisfies \(\widehat{\phi}(2\xi)=m_0(\xi)\widehat{\phi}(\xi)\) for some periodic \(m\)) is said to satisfy the unitary extension principle if, whenever the matrix \(M(\xi)\) satisfies \(M^\ast(\xi)M(\xi)=I_2\) then the affine system (\(\psi_{j,k}=2^{j/2}\psi(2^j x-k)\)) generated by \(\{\phi,\psi_1,\dots,\psi_r\}\) with \(\widehat{\psi}_k(2\xi)=m_k(\xi)\widehat{\phi}(\xi)\), defines a Parseval wavelet frame for \(L^2(\mathbb{R})\).
Here the authors prove the following about \(\phi\) satisfying the unitary extension principle. Suppose that \(\mathcal{M}=\{m_0,m_1,\dots,m_r\}\) and \(\mathcal{N}=\{m_0,n_1,\dots,n_r\}\) are such that \(M^\ast M(\xi)=I_2=N^\ast N(\xi)\) and \(\widetilde{M}^\ast(\xi)\widetilde{N}(\xi)=0\) for almost all \(\xi\in [0,1)\). Then the affine systems generated by \(\psi_k\) and \(\eta_k\) where \(\widehat{\psi}_k(2\xi)=m_k(\xi)\widehat{\psi}_k(\xi)\) and \(\widehat{\eta}_k(2\xi)=n_k(\xi)\widehat{\eta}_k(\xi)\) form orthogonal Parseval wavelet frames. A more general construction of pairwise orthogonal multiwavelet frames is also provided and applied in higher dimensions.
The purpose of these results is to define true vector-valued discrete wavelet transforms in which wavelet expansions do not reduce to coordinatewise expansions. Discrete implementations and applications to wavelet decomposition of color images are outlined.

MSC:
42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46E20 Hilbert spaces of continuous, differentiable or analytic functions
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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