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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 $$L^2 (\mathbb{R}^d)$$.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
##### Keywords:
biorthogonal wavelet bases; Wiener class; scaling functions
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