BIORTHOGONAL WAVELET BASES ON R(D)

The paper investigates the construction of biorthogonal wavelet bases on R(d). Assume that M(xi)(M) over tilde(t)(xi) = I for all xi is an e lement of T-d, where M(xi) = (m(mu)(xi + nu pi))(mu nu is an element o f E), (M) over tilde(xi) = ((m) over tilde,(mu)(xi + nu pi))(mu,nu is an element of E) with all m(mu)(xi) and (m) over tilde(mu)(xi) (mu is an element of E) being in the Wiener class W(T-d). Let phi and <(phi)o ver tilde> be the associated scaling functions, {psi(mu} and {psi?(mu) }(mu is an element of E - {0}) be the associated wavelet functions. Un der 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) implie s (3); (1) implies (4); and in the case when m(0)(xi) and (m) over til de(0)() are trigonometric polynomials, (4) implies (1);and (5). These five assertions are: (1) Phi(xi) approximate to 1 approximate to 1 (Ph i(xi)(Phi(xi) = Sigma(alpha) \<(phi)over cap>(xi + 2 alpha pi)\(2)); ( 2) [phi, <(phi)over tilde>(. - k)] = delta(0,k); (3) [(psi mu,j,k') ]d elta>ir',j',k') = delta(mu mu')delta(jj')delta(kk'); (4) \lambda\(max) < 1, <(lambda)over tilde>\(max) < 1 (lambda's are eigenvalues of tran sition operators restricted on P-0); (5) {(psi mu,j,k), (<(psi)over) ( tilde>mu,j,k)} is a dual Riesz basis of L(2)(R(d)). (C) 1995 Academic Press, Inc.