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Vector-Valued Wavelet Bases as Hilbert 𝕄m(ℝ)-Module Bases: A Construction from Scalar Wavelets

  • Hicham Tarif
  • Nadir Maaroufi
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Abstract

Vector-valued multiscale representations are essential when signals or fields take values in ℝm 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 L2(ℝd,ℝm) by endowing this space with a natural 𝕄m(ℝ)-valued inner product, thereby turning it into a Hilbert 𝕄m(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 L2(ℝd,ℝm) from scalar wavelet bases while preserving compact support, vanishing moments, and regularity.

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Cite

BibTeXRIS
BibTeX
@misc{tarif2026vectorvalued,
  title = {{Vector-Valued Wavelet Bases as Hilbert $\mathbbM_m(\mathbbR)$-Module Bases: A Construction from Scalar Wavelets}},
  author = {Hicham Tarif and Nadir Maaroufi},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30589},
  eprint = {2608.30589},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30589},
}
RIS
TY  - UNPB
TI  - Vector-Valued Wavelet Bases as Hilbert 𝕄_m(ℝ)-Module Bases: A Construction from Scalar Wavelets
AU  - Hicham Tarif
AU  - Nadir Maaroufi
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30589
UR  - https://arxiv.org/abs/2608.30589
AB  - Vector-valued multiscale representations are essential when signals or fields take values in 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(ℝ^d,ℝ^m) by endowing this space with a natural 𝕄_m(ℝ)-valued inner product, thereby turning it into a Hilbert 𝕄_m(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(ℝ^d,ℝ^m) from scalar wavelet bases while preserving compact support, vanishing moments, and regularity.
ER  -