Wilson-Daubechies-Meyer Bases Overview

• Wilson-Daubechies-Meyer bases are orthonormal bases in L²(ℝᵈ) constructed from structured time-frequency shifts, generalizing classical Gabor systems.
• They employ redundancy and precise linear combinations to achieve superior time-frequency localization while bypassing the Balian–Low theorem in one dimension.
• Multidimensional extensions use group theory and tight Gabor frames to ensure robust modular tiling of phase space with well-localized atoms.

A Wilson–Daubechies–Meyer (WDM) basis is an orthonormal basis for $L^2(\mathbb{R}^d)$ constructed from finite linear combinations of time–frequency shifts (also known as Weyl–Heisenberg operators) of a square-integrable function. These bases generalize classical Gabor systems by circumventing the Balian–Low theorem’s obstruction to simultaneously well-localized and orthonormal Gabor bases in dimension one, via a systematic use of redundancy and structured linear combinations of time–frequency atoms. The foundational WDM constructions admit multidimensional generalizations corresponding to higher redundancies and separability properties, yielding bases with strong time–frequency localization and modular tilings of phase space (Bownik et al., 2017).

1. Time–Frequency Shifts, Gabor Frames, and the Balian–Low Obstruction

Time–frequency shifts act on $L^2(\mathbb{R}^d)$ via translation $T_x f(t) = f(t-x)$ and modulation $$M_\omega f(t) = e^{2\pi i \langle \omega, t \rangle} f(t)$$, combined as $\pi(x, \omega) = M_\omega T_x$. Gabor systems $$\mathcal{G}(g; A, \Lambda)=\{\pi(a, \lambda) g: a\in A,\lambda\in\Lambda\}$$ are frames when the window $g$ and the lattices $A,\Lambda \subset \mathbb{R}^d$ satisfy specific tightness and redundancy constraints. The redundancy, $(|\det A| \cdot |\det \Lambda|)^{-1}$, quantifies the oversampling in time–frequency space.

The Balian–Low theorem asserts that in $d=1$, no orthonormal Gabor basis on a redundancy-1 lattice can use a window $g\in L^2(\mathbb{R})$ that is well localized in both time and frequency, thereby motivating alternative constructions like Wilson systems for such localization.

2. Construction of One-Dimensional Wilson Systems

A one-dimensional Wilson system is derived from a redundancy-2, half-shifted Gabor system generated by an even and normalized $g$:

• $g(x) = g(-x)$,
• $\widehat{g}(\omega) = \widehat{g}(-\omega)$,
• $\|g\|_2 = 1$,
• the half-shift Gabor frame $\{M_n T_{m/2} g\}_{m,n \in \mathbb{Z}}$ is tight with frame bound 2.

The Wilson system comprises three atom families:

• Pure translations: $\psi_{0,n}(x) = T_n g(x)$,
• Even-modulation combinations (for $m>0$): $$\psi_{m,n}(x) = \frac{1}{\sqrt{2}} T_n\{ M_m + (-1)^n M_{-m} \}g(x)$$,
• Odd-modulation combinations (for $m>0$): $$\varphi_{m,n}(x) = \frac{1}{i\sqrt{2}} T_{n+1/2} \{ M_m - (-1)^n M_{-m} \}g(x)$$.

Daubechies–Jaffard–Journé’s theorem establishes that $W(g)$ is an orthonormal basis for $L^2(\mathbb{R})$ if and only if the generating Gabor frame is tight of bound 2.

3. Characterization of Tightness Criteria and Autocorrelation Conditions

The tightness condition for the underlying Gabor system can be expressed in the Fourier domain via autocorrelation functions. For the half-shifted redundancy-2 Gabor frame in one dimension, the requirement is:

$$\sum_{m \in \mathbb{Z}} |\widehat{g}(\omega - m)|^2 = 2 \quad \text{a.e. }\omega \in \mathbb{R}.$$

This ensures that the Wilson system built from such $g$ is orthonormal and complete. In $d$ dimensions, generalizations use subgroups $G \subset \mathbb{Z}^d/(2\mathbb{Z}^d)$ of order $2^k$, leading to conditions of the form:

$$\sum_{y \in A} |\widehat{g}(\omega - y)|^2 = 2^k, \quad \text{a.e.} \ \omega \in \mathbb{R}^d,$$

combined with evenness of $g$ in each coordinate.

4. Localization and Generator Properties

The generator $\psi$ (identically $g$ in the one-dimensional WDM context) can be selected from the Schwartz class $\mathcal{S}(\mathbb{R})$ or the Feichtinger algebra $S_0(\mathbb{R})$, so that the resultant basis elements exhibit rapid decay in both domains. For all $N, M \geq 0$, constants $C_{N,M}$ and $C'_{N,M}$ satisfy: $$|x^N D^M \psi_{n,k}(x)| \leq C_{N,M}(1+|x-k|)^{-N}, \qquad |\omega^N D^M \widehat{\psi}_{n,k}(\omega)| \leq C'_{N,M}(1+|\omega \mp n|)^{-N},$$ demonstrating time–frequency localization and concentration of atoms around specified time and frequency indices.

5. Multidimensional Wilson Bases and Redundancy-$2^k$ Gabor Frames

With $d \geq 1$ and a separable subgroup $G \subset \mathbb{Z}^d/(2\mathbb{Z}^d)$ of order $2^k$, multidimensional WDM bases are constructed via:

• Generator $g \in L^2(\mathbb{R}^d)$ even in each coordinate of time and frequency,
• $g$ is $G$-separable, written as a product across disjoint coordinate subsets,
• The Gabor frame $\{\pi(a/2, n) g : a \in A, n \in \mathbb{Z}^d\}$ with $A = \cup_{o \in G}(o+2\mathbb{Z}^d)$ is tight of bound $2^k$.

Wilson system elements are of the form $T_m \Phi_{h,y}(x)$, where each generator

$$\Phi_{h,y}(x) = 2^{-k/2} \sum_{\sigma \in G} (-1)^{\langle I(h) + y, \sigma \rangle} M_{R_\sigma y} g(x)$$

involves all group elements via reflection operators $R_o$ and dual pairing $\langle \cdot, \cdot \rangle$ modulo 2, with $I$ the canonical isomorphism $G \to G^{\vee}$. This yields an orthonormal basis for $L^2(\mathbb{R}^d)$, with each atom covering a $2^k$-modular region in time–frequency space and possessing $2^k$ spectral peaks, one per coset of $G$.

6. Foundational Theorems and Structural Arguments

The following outcomes encapsulate existence and structural results:

• In $d=1$, a tight Gabor frame of redundancy 2 with even window yields a Wilson ONB precisely (Theorem 1.2).
• For higher $d$ and $k$, a "tower" of orthonormal Wilson bases (one for each $G$ of order $2^k$) can be constructed with tight Gabor frames of suitable redundancy (Theorem 5.1).
• If $g$ is chosen smooth and rapidly decaying (Schwartz or Feichtinger), localization of WDM atoms persists, surpassing the Balian–Low obstruction when redundancy exceeds one.

All proofs are anchored in shift-invariant system theory, via Ron–Shen dual Gramian/autocorrelation analysis, and a commutator lemma ensuring pairwise orthogonality of Wilson atoms in the tight-frame case. The Wilson system thus obtained forms a Parseval frame with unit-norm generators and is, by construction, an orthonormal basis.

For a comprehensive technical development, see Bownik, Jakobsen, Lemvig, and Okoudjou (Bownik et al., 2017).