Continuous Algebraic Diversity: Unifying Spectral, Wavelet, and Time-Frequency Analysis via Lie Group Actions
Fuente:
arXiv
Enregistré dans:
| Auteur principal: | |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911639610064896 |
|---|---|
| author | Thornton, Mitchell A. |
| author_facet | Thornton, Mitchell A. |
| contents | We provide a computable criterion for selecting among Fourier, wavelet, and time-frequency analysis by extending the algebraic diversity (AD) framework to Lie groups acting on $L^2(\mathbb{R})$. To our knowledge, there is no other criterion that provides this selection capability. The group-averaged estimator generalizes from a finite sum over group elements to an integral with respect to Haar measure. A Continuous Replacement Theorem establishes signal-noise separation under equivariance and ergodicity conditions, with a noise operator $\mathcal{N}_G = C_ρ^{-2}$ determined by the Duflo-Moore operator that explains the frequency-dependent noise floor in wavelet analysis as a consequence of the affine group's non-unimodularity. A Unification Theorem shows that classical spectral analysis corresponds to the translation group, wavelet analysis to the affine group, time-frequency analysis to the Heisenberg-Weyl group, and spherical harmonics to SO(3). The commutativity residual $δ$, extended to Hilbert-Schmidt operator norms, provides a principled selection criterion among these groups. A double-commutator generalized eigenvalue problem solves the blind group matching problem in polynomial time. A Discretization Recovery Theorem establishes that all discrete AD results are sampling approximations to the continuous theory, with $\mathbb{Z}_M \to (\mathbb{R},+)$ as $M \to \infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00848 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Continuous Algebraic Diversity: Unifying Spectral, Wavelet, and Time-Frequency Analysis via Lie Group Actions Thornton, Mitchell A. Signal Processing Information Theory Functional Analysis We provide a computable criterion for selecting among Fourier, wavelet, and time-frequency analysis by extending the algebraic diversity (AD) framework to Lie groups acting on $L^2(\mathbb{R})$. To our knowledge, there is no other criterion that provides this selection capability. The group-averaged estimator generalizes from a finite sum over group elements to an integral with respect to Haar measure. A Continuous Replacement Theorem establishes signal-noise separation under equivariance and ergodicity conditions, with a noise operator $\mathcal{N}_G = C_ρ^{-2}$ determined by the Duflo-Moore operator that explains the frequency-dependent noise floor in wavelet analysis as a consequence of the affine group's non-unimodularity. A Unification Theorem shows that classical spectral analysis corresponds to the translation group, wavelet analysis to the affine group, time-frequency analysis to the Heisenberg-Weyl group, and spherical harmonics to SO(3). The commutativity residual $δ$, extended to Hilbert-Schmidt operator norms, provides a principled selection criterion among these groups. A double-commutator generalized eigenvalue problem solves the blind group matching problem in polynomial time. A Discretization Recovery Theorem establishes that all discrete AD results are sampling approximations to the continuous theory, with $\mathbb{Z}_M \to (\mathbb{R},+)$ as $M \to \infty$. |
| title | Continuous Algebraic Diversity: Unifying Spectral, Wavelet, and Time-Frequency Analysis via Lie Group Actions |
| topic | Signal Processing Information Theory Functional Analysis |
| url | https://arxiv.org/abs/2605.00848 |