Biorthogonal Wavelet Bases on [formula omitted

Abstract

The paper investigates the construction of biorthogonal wavelet bases on R d . Assume that M(ξ)[formula](ξ) = I for all ξ ∈ T d , where M(ξ) = ( m μ(ξ + νπ)) μ,ν∈ E , M̃(ξ) = ( m̃ μ(ξ + νπ )) μ,ν∈ E with all m μ(ξ) and m̃ μ(ξ) (μ ∈ E) being in the Wiener class W( T d ). Let φ and φ̃ be the associated scaling functions, {ψ μ} and {ψ̃ μ}(μ ∈ 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(ξ) and m̃ 0(ξ) are trigonometric polynomials, (4) implies (1) and (5). These five assertions are: (1) Φ(ξ) ≍ 1 ≍ Φ̃(ξ)(Φ(ξ) = ∑ α |φ̂(ξ + 2απ)| 2); (2) 〈φ, φ̃(· - k)〉 = δ 0, k ; (3) 〈ψ μ, j, k ,ψ̃ μ′, j′, k′ 〉 = δ μμ′δ jj′ δ kk′ ; (4) |λ| max < 1, |λ̃| max < 1 (λ′s are eigenvalues of transition operators restricted on P 0); (5) {ψ μ, j, k ,ψ̃ μ, j, k } is a dual Riesz basis of L 2( R d ).