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Biorthogonal wavelet bases on \(\mathbb{R}^ d\). (English) Zbl 0846.42018
Summary: The paper investigates the construction of biorthogonal wavelet bases on \(\mathbb{R}^d\). Assume that \(M(\xi) \overline {\widetilde {M}^t} (\xi) =I\) for all \(\xi\in T^d\), where \(M(\xi)= (m_\mu (\xi+ \nu\pi ))_{\mu, \nu\in E}\), \(\widetilde {M} (\xi)= (\widetilde {m}_\mu (\xi+ \nu\pi ))_{\mu, \nu\in E}\) with all \(m_\mu (\xi)\) and \(\widetilde {m}_\mu (\xi)\) \((\mu\in E)\) being in the Wiener class \(W(T^d)\). Let \(\varphi\) and \(\widetilde {\varphi}\) be the associated scaling functions, \(\{\psi_\mu\}\) and \(\{\widetilde {\psi}_\mu \}\) \((\mu\in E- \{0\})\) be the associated wavelet functions. Under weaker conditions and with simpler proofs, this paper obtains the following results: (1) and (2) are equivalent; (2) implies (3) always, and (3) implies (2) under some additional mild conditions; (5) implies (3); (1) implies (4); and in the case when \(m_0 (\xi)\) and \(\widetilde {m}_0 (\xi)\) are trigonometric polynomials, (4) implies (1) and (5). These five assertions are: (1) \(\Phi (\xi) \approx 1\approx \widetilde {\Phi} (\xi)\) \((\Phi (\xi)= \sum_\alpha |\widehat {\varphi} (\xi+ 2\alpha\pi) |^2)\); (2) \(\langle \varphi, \widetilde {\varphi} (\cdot- k)\rangle= \delta_{0,k}\); (3) \(\langle \psi_{\mu,j, k}, \widetilde {\psi}_{\mu', j', k'} \rangle= \delta_{\mu \mu'} \delta_{jj'} \delta_{kk'}\); (4) \(|\lambda |_{\max}<1\), \(|\widetilde {\lambda} |_{\max} <1\) (\(\lambda\)’s are eigenvalues of transition operators restricted on \({\mathcal P}_0)\); (5) \(\{\psi_{\mu, j, k}, \widetilde {\psi}_{\mu, j, k}\}\) is a dual Riesz basis of \(L^2 (\mathbb{R}^d)\).

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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