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

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.