# Symmetric orthonormal complex wavelets with masks of arbitrarily high linear-phase moments and sum rules

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## Abstract

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 2π-periodic trigonometric polynomial $$\hat a$$ with complex coefficients such that

1. (i)

$$\hat a$$ is an orthogonal mask: $$|\hat a(\xi)|^2+|\hat a(\xi+\pi)|^2=1$$.

2. (ii)

$$\hat a$$ has m + 1 − odd m sum rules: $$\hat a(\xi+\pi)=O(|\xi|^{m+1-odd_m})$$ as ξ→0, where $$odd_m:=\frac{1-(-1)^m}{2}$$.

3. (iii)

$$\hat a$$ has m + odd m linear-phase moments: $$\hat a(\xi)=e^{{{\mathrm{i}}} c\xi}+O(|\xi|^{m+odd_m})$$ as ξ→0 with phase c = − 1/2.

4. (iv)

$$\hat a$$ has symmetry and coefficient support [2 − 2m,2m − 1]: $$\hat a(\xi)=\sum_{k=2-2m}^{2m-1} h_k e^{-{{\mathrm{i}}} k\xi}$$ with h1 − k  = h k .

5. (v)

$$\hat a(\xi)\ne 0$$ for all ξ ∈ ( − π,π).

Define $$\hat \phi(\xi):=\prod_{j=1}^\infty \hat a(2^{-j}\xi)$$ and $$\hat \psi(2\xi)=e^{-{{\mathrm{i}}} \xi} {\overline{\hat a(\xi+\pi)}}\hat \phi(\xi)$$. 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: $$\int_{{{\mathbb R}}} \phi(x)dx=1$$ and $$\int_{{{\mathbb R}}} (x-1/2)^j \phi(x) dx=0$$ for all j = 1,...,m + odd m  − 1. Both functions ϕ and ψ are supported on [2 − 2m,2m − 1].

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## Author information

Authors

### Corresponding author

Correspondence to Bin Han.

Communicated by Qiyu Sun.

Research supported in part by NSERC Canada under Grant RGP 228051.

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Han, B. Symmetric orthonormal complex wavelets with masks of arbitrarily high linear-phase moments and sum rules. Adv Comput Math 32, 209–237 (2010). https://doi.org/10.1007/s10444-008-9102-7

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• DOI: https://doi.org/10.1007/s10444-008-9102-7