向量值小波基作为希尔伯特$\boldsymbol{\rm M}_m(\boldsymbol{\rm R})$-模基：从标量小波的构造
Vector-Valued Wavelet Bases as Hilbert $\mathbb{M}_m(\mathbb{R})$-Module Bases: A Construction from Scalar Wavelets
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中文总结 AI 辅助
该研究针对取值于$\boldsymbol{\rm R}^m$的信号，构建希尔伯特$\boldsymbol{\rm M}_m(\boldsymbol{\rm R})$-模框架下的向量值小波基，通过从标量小波的构造性提升过程，保留紧支集等性质并编码分量交互。
中文摘要 AI 辅助
当信号或场取值于$\boldsymbol{\rm R}^m$且分量间交互携带有效信息时，向量值多尺度表示至关重要。多数多小波与超小波构造在标量希尔伯特空间框架下展开，通常生成逐通道标量系数后再进行重组。我们为$L^2(\boldsymbol{\rm R}^d,\boldsymbol{\rm R}^m)$上的向量值小波开发了一个内在框架，通过为该空间赋予自然的$\boldsymbol{\rm M}_m(\boldsymbol{\rm R})$-值内积，将其转化为希尔伯特$\boldsymbol{\rm M}_m(\boldsymbol{\rm R})$-模。该模视角产生编码分量间交互的矩阵值系数，并通过帕塞瓦尔型恒等式提供规范重构。在该框架内，我们引入一种构造性提升过程，从标量小波基出发，在$L^2(\boldsymbol{\rm R}^d,\boldsymbol{\rm R}^m)$中构建可分多变量向量值小波基，同时保持紧支集、消失矩和正则性。
英文摘要
Vector-valued multiscale representations are essential when signals or fields take values in $\mathbb{R}^m$ and component interactions carry meaningful information. Most multiwavelet and super-wavelet constructions are formulated in scalar Hilbert-space settings and typically produce channelwise scalar coefficients followed by recombination. We develop an intrinsic framework for vector-valued wavelets on $L^2(\mathbb{R}^d,\mathbb{R}^m)$ by endowing this space with a natural $\mathbb{M}_m(\mathbb{R})$-valued inner product, thereby turning it into a Hilbert $\mathbb{M}_m(\mathbb{R})$-module. This module viewpoint yields matrix-valued coefficients that encode cross-component interactions and provides canonical reconstruction through a Parseval-type identity. Within this setting, we introduce a constructive lifting procedure that builds separable multivariate vector-valued wavelet bases in $L^2(\mathbb{R}^d,\mathbb{R}^m)$ from scalar wavelet bases while preserving compact support, vanishing moments, and regularity.
发表机构
- International University of Rabat（拉巴特国际大学）
机构由 AI 辅助整理，请以论文原文为准。
