In this paper, we investigate compactly supported symmetric orthonormal dyadic complex wavelets such that the symmetric orthonormal
refinable functions have high linear-phase moments and the antisymmetric wavelets have high vanishing moments. Such wavelets
naturally lead to real-valued symmetric tight wavelet frames with some desirable moment properties, and are related to coiflets
which are real-valued and are of interest in numerical algorithms. For any positive integer
m, employing only the Riesz lemma without solving any nonlinear equations, we obtain a 2
π-periodic trigonometric polynomial
[^(a)]\hat a with complex coefficients such that
| (i) |
[^(a)]\hat a is an orthogonal mask: |[^(a)](x)|2+|[^(a)](x+p)|2=1|\hat a(\xi)|^2+|\hat a(\xi+\pi)|^2=1.
|
| (ii) |
[^(a)]\hat a has m + 1 − odd
m
sum rules: [^(a)](x+p)=O(|x|m+1-oddm)\hat a(\xi+\pi)=O(|\xi|^{m+1-odd_m}) as ξ→0, where
oddm:=\frac1-(-1)m2odd_m:=\frac{1-(-1)^m}{2}.
|
| (iii) |
[^(a)]\hat a has m + odd
m
linear-phase moments: [^(a)](x)=ei cx+O(|x|m+oddm)\hat a(\xi)=e^{{{\mathrm{i}}} c\xi}+O(|\xi|^{m+odd_m}) as ξ→0 with phase c = − 1/2.
|
| (iv) |
[^(a)]\hat a has symmetry and coefficient support [2 − 2m,2m − 1]: [^(a)](x)=åk=2-2m2m-1 hk e-i kx\hat a(\xi)=\sum_{k=2-2m}^{2m-1} h_k e^{-{{\mathrm{i}}} k\xi} with h
1 − k
= h
k
.
|
| (v) |
[^(a)](x) ¹ 0\hat a(\xi)\ne 0 for all ξ ∈ ( − π,π).
|
Define
[^(f)](x):=Õj=1¥ [^(a)](2-jx)\hat \phi(\xi):=\prod_{j=1}^\infty \hat a(2^{-j}\xi) and
[^(y)](2x)=e-i x [`([^(a)](x+p))][^(f)](x)\hat \psi(2\xi)=e^{-{{\mathrm{i}}} \xi} {\overline{\hat a(\xi+\pi)}}\hat \phi(\xi). Then
ψ is a compactly supported antisymmetric orthonormal wavelet with
m + 1 −
odd
m
vanishing moments, and
ϕ is a compactly supported symmetric orthonormal refinable function with the special linear-phase moments:
ò\mathbb R f(x)dx=1\int_{{{\mathbb R}}} \phi(x)dx=1 and
ò\mathbb R (x-1/2)j f(x) dx=0\int_{{{\mathbb R}}} (x-1/2)^j \phi(x) dx=0 for all
j = 1,...,
m +
odd
m
− 1. Both functions
ϕ and
ψ are supported on [2 − 2
m,2
m − 1].
The mask of a coiflet has real coefficients and satisfies (i), (ii), and (iii), often with a general phase c and the additional condition that the order of the linear-phase moments is equal (or close) to the order of the sum rules.
On the one hand, as Daubechies showed in [3, 5] that except the Haar wavelet, any compactly supported dyadic orthonormal real-valued wavelets including coiflets cannot
have symmetry. On the other hand, solving nonlinear equations, [4, 12] constructed many interesting real-valued dyadic coiflets without symmetry. But it remains open whether there is a family
of real-valued orthonormal wavelets such as coiflets whose masks can have arbitrarily high linear-phase moments. This partially
motivates this paper to study the complex wavelet case with symmetry property. Though symmetry can be achieved by considering
complex wavelets, the symmetric Daubechies complex orthogonal masks in [11] generally have no more than 2 linear-phase moments. In this paper, we shall study and construct orthonormal dyadic complex
wavelets and masks with symmetry, linear-phase moments, and sum rules. Examples and two general construction procedures for
symmetric orthogonal masks with high linear-phase moments and sum rules are given to illustrate the results in this paper.
We also answer an open question on construction of symmetric Daubechies complex orthogonal masks in the literature.
Keywords Orthonormal complex wavelets - Symmetry - Linear-phase moments - Sum rules - Vanishing moments
Mathematics Subject Classifications (2000) 42C40 - 42C05 - 41A25
Communicated by Qiyu Sun.
Research supported in part by NSERC Canada under Grant RGP 228051.