Biorthogonal Wavelet Bases on Rd

✍ Scribed by Ruilin Long; Dirong Chen

• Publisher: Elsevier Science
• Year: 1995
• Tongue: English
• Weight: 714 KB
• Volume: 2
• Category: Article
• ISSN: 1063-5203
• DOI: 10.1006/acha.1995.1016

✦ Synopsis

The paper investigates the construction of biorthogonal wavelet bases on (\mathbb{R}^{d}). Assume that (M(\xi) \tilde{M}^{\prime}(\xi)=I) for all (\xi \in T^{d}), where (M(\xi)=\left(m_{\mu}(\xi+\nu \pi)\right){\mu, \nu \in E E}, \tilde{M}(\xi)=\left(\tilde{m}{\mu}(\xi+\nu \pi)\right){\mu, \nu \in E}) with all (m{\mu}(\xi)) and (\tilde{m}{\mu}(\xi)(\mu \in E)) being in the Wiener class (W\left(T^{d}\right)). Let (\varphi) and (\tilde{\varphi}) be the associated scaling functions, (\left{\psi{\mu}\right}) and (\left{\tilde{\psi}{\mu}\right}(\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 (\tilde{m}{0}(\xi)) are trigonometric polynomials, (4) implies (1) and (5). These five assertions are: ((1) \Phi(\xi) \approx 1 \approx \tilde{\Phi}(\xi)\left(\Phi(\xi)=\sum{\alpha}|\hat{\varphi}(\xi+2 \alpha \pi)|^{2}\right) ;(2)) (\langle\varphi, \tilde{\varphi}(\cdot k)\rangle=\delta_{0, k} ;) (3) (\left\langle\psi_{\mu, j, k}, \tilde{\psi}{\mu^{\prime}, j^{\prime}, k^{\prime}}\right\rangle=\delta{\mu \mu^{\prime}} \delta_{j j^{\prime}} \delta_{k k^{\prime}} ;) (4) (|\lambda|{\max }<) (1,|\tilde{\lambda}|{\max }<1(\lambda) 's are eigenvalues of transition operators restricted on (\left.\mathscr{P}{0}\right) ;(5)\left{\psi{\mu, j, k}, \tilde{\psi}_{\mu, j, k}\right}) is a dual Riesz basis of (L^{2}\left(\mathbb{R}^{d}\right)). (c) 1995