A Constructive Approach for Building Wavelet Bases in \( L^2(\mathbb{R}^d, \mathbb{R}^m) \) with Optimal Properties

Abstract

The main contribution of this paper is a constructive method for building separable multivariate vector-valued wavelet bases in the general framework of \( L^2(\mathbb{R}^d, \mathbb{R}^m) \) for any \( d, m \geq 1 \). While separable wavelet bases in \( L^2(\mathbb{R}^d, \mathbb{R}) \) are well-established and widely applied, the explicit construction of truly vector-valued wavelet bases remains an open problem, even in the simplest case of \( L^2(\mathbb{R}, \mathbb{R}^2) \), let alone in \( L^2(\mathbb{R}^2, \mathbb{R}^2) \). In practice, the conventional approach applies standard separable wavelet bases of \( L^2(\mathbb{R}^2, \mathbb{R}) \) independently to each component of vector-valued signals in \( L^2(\mathbb{R}^2, \mathbb{R}^2) \). However, this approach fails to capture the intrinsic vectorial structure of the signals. To address this limitation, we propose a constructive approach within the vector-valued wavelet framework, providing a systematic method for constructing such bases in the general case of \( L^2(\mathbb{R}^d, \mathbb{R}^m) \). By linking \( m \)-multiwavelets to vector-valued wavelets, our approach not only enables the systematic construction of separable multivariate bases in \( L^2(\mathbb{R}^d, \mathbb{R}^m) \) that satisfy the vector-valued multiresolution analysis but also ensures that these bases inherit key structural properties, making them well-suited for practical applications.