A Constructive Approach for Building Wavelet Bases in \( L^2(\mathbb{R}^d, \mathbb{R}^m) \) with Optimal Properties
Hicham Tarif, Nadir Maaroufi
TL;DR
This work addresses the explicit construction of vector-valued wavelet bases in $L^2(\mathbb{R}^d,\mathbb{R}^m)$ by introducing a constructive framework that directly builds vector-valued scaling and mother wavelets from univariate scalar bases, and by connecting $m$-multiwavelets to vector-valued wavelets through the VMRA setting. It then extends this one-dimensional construction to multivariate spaces via a tensor-product approach, leveraging $m$-multiwavelets to obtain separable vector-valued bases in $L^2(\mathbb{R}^d,\mathbb{R}^m)$ that preserve compact support, regularity, and vanishing moments. A key result provides necessary and sufficient conditions for orthonormal vector-valued wavelets in $L^2(\mathbb{R},\mathbb{R}^m)$ and demonstrates how to assemble multivariate bases by tensoring these components, with explicit constructions illustrated for $d=m=2$. The framework yields VMRA-compliant, structure-preserving bases capable of capturing inter-component correlations, offering theoretical and practical benefits for vector-valued signal analysis in fluid dynamics, multichannel processing, and related fields. The work also outlines future directions toward non-separable vector-valued wavelets and potential integration with deep learning to enhance interpretability and feature extraction in multivariate data analysis.
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.