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

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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].

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.

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Correspondence to Bin Han.

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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).

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