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

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 â with complex coefficients such that (i) â is an orthogonal mask: |â(ξ)|2 + |â(ξ + π)|2 = 1. (ii) â has m + 1− oddm sum rules: â(ξ + π) = O(|ξ|m+1−oddm) as ξ → 0, where oddm := 1−(−1) m