Construction of a class of multivariate compactly supported wavelet bases for L^{\mathrm{2}}({ℝ}^d)

Abstract

In this paper, for a given d×d expansive matrix M with |det M| = 2, we investigate the compactly supported M-wavelets for L^{\mathrm{2}}({ℝ}^d). Starting with a pair of compactly supported refinable functions \varphi and \tilde{\varphi } satisfying a mild condition, we obtain an explicit construction of a compactly supported wavelet ψ such that \{{\mathrm{2}}^{j/\mathrm{2}}\psi (M^j·-k):j\in \mathbb{Z},k\in {\mathbb{Z}}^d\} forms a Riesz basis for L^{\mathrm{2}}({ℝ}^d). The (anti-)symmetry of such ψ is studied, and some examples are also provided.

Key words： Riesz basis; wavelet; refinable function

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