A general theory of alternative-norm dual Gabor windows: complementary operators for arbitrary regularizers and non-quadratic optimality criteria

For a Gabor (Weyl–Heisenberg) frame in $L^2(\mathbb{R})$, the canonical dual window minimizes the $L^2$-norm among all dual windows, but it may lack desirable time-frequency concentration, support, or smoothness. Daubechies, H. J. Landau, and Z. Landau (J. Fourier Anal. Appl. 1 (1994) 437–478) constructed, in Section 7 of their study of the Wexler–Raz identity, dual windows minimizing a quadratic, operator-weighted norm $\|\Lambda g^b\|$: one introduces a complementary operator $\Omega$ on $\ell^2(\mathbb{Z}^2)$, assumes a two-sided norm equivalence, and obtains the minimizer in closed form provided the operator $\Omega\,T_{g;1/\beta,1/\alpha}\Lambda^{-2}T^*_{g;1/\beta,1/\alpha}\,\Omega$ is invertible on $\ell^2$. Sufficient conditions were given only for specific regularizers $\Lambda$ (e.g. diagonal weights, multiplication or differential-type operators). A substantial later literature treats alternative dual windows only in special regimes — sparse and minimal-support duals, numerically computed convex-optimal duals, and $L^1$-minimal duals in particular cases — while the general questions remain open: which complementary operators $\Omega$ are admissible for an arbitrary regularizer $\Lambda$ (with sharp two-sided bounds), what are necessary and sufficient conditions for the invertibility underlying the closed-form minimizer, and when do duals optimizing general convex (non-quadratic) criteria exist, are unique, and are stable. The related request for general truncation-error/complexity bounds is likewise only partially answered.

Background

A Gabor system (also called a Weyl–Heisenberg system) is the family of time-frequency shifts $g_{m\alpha,n\beta}(t)=e^{-2\pi i m\alpha t}g(t-n\beta)$, $m,n\in\mathbb{Z}$, built from a single window $g\in L^2(\mathbb{R})$ on the lattice $(m\alpha,n\beta)$. Such a system is a frame for $L^2(\mathbb{R})$ — meaning $A\|f\|_2^2\le\sum_{m,n}|\langle f,g_{m\alpha,n\beta}\rangle|^2\le B\|f\|_2^2$ for constants $0<A\le B$ — precisely when every $f$ can be stably reconstructed from the inner products $\langle f,g_{m\alpha,n\beta}\rangle$. Reconstruction uses a dual window $h$ with $f=\sum_{m,n}\langle f,g_{m\alpha,n\beta}\rangle h_{m\alpha,n\beta}$; the canonical dual $\tilde g$ (obtained by applying the inverse frame operator to $g$) is the dual of minimal $L^2$-norm. Wexler and Raz (1990), studying discrete Gabor expansions, had observed that duality of windows can be characterized on the reciprocal (dual) lattice $(m/\beta,n/\alpha)$; Daubechies, H. J. Landau, and Z. Landau (1994) established this Wexler–Raz identity rigorously at the operator level. Writing $T_{g;\alpha,\beta}:L^2\to\ell^2(\mathbb{Z}^2)$ for the analysis operator $T_{g;\alpha,\beta}f=(\langle f,g_{m\alpha,n\beta}\rangle)_{m,n}$, the identity reads $T^*_{f;\alpha,\beta}T_{g;\alpha,\beta}h=\frac{1}{\alpha\beta}T^*_{h;1/\beta,1/\alpha}T_{g;1/\beta,1/\alpha}f$, and it implies that dual windows $g^b$ are exactly the solutions of the biorthogonality-type equation $T_{g;1/\beta,1/\alpha}\,g^b=\alpha\beta\,e_{0,0}$ on the reciprocal lattice, with $e_{0,0}$ the Kronecker sequence. The same paper recast this through the theory of von Neumann algebras (algebras of bounded operators closed under adjoints and weak limits), where the commuting algebras generated by the time-frequency shifts on the two lattices are related by a fundamental invariant, the coupling constant, equal to $1/(\alpha\beta)$ here; this gave an elementary proof that a spanning Gabor system needs $\alpha\beta\le1$ (unit lattice density), and Neumann-series type iterations (iterative summation of an operator geometric series $\sum_{k\ge0}A^k$) for computing duals with geometrically convergent error. In Section 7 the authors went beyond minimal-$L^2$ duals: given a positive operator $\Lambda$ (a regularizer modeling smoothness or concentration, e.g. multiplication by a weight or a differential-type operator), they sought the dual window $g^b$ minimizing $\|\Lambda g^b\|$ subject to the Wexler–Raz constraint. Their recipe introduces a complementary operator $\Omega$ on $\ell^2(\mathbb{Z}^2)$ and requires a two-sided inequality $A'\|\Omega^{-1}c\|^2\le\|\Lambda^{-1}T^*_{g;1/\beta,1/\alpha}c\|^2\le B'\|\Omega^{-1}c\|^2$ for all $c\in\ell^2$; when the operator $\Omega\,T_{g;1/\beta,1/\alpha}\Lambda^{-2}T^*_{g;1/\beta,1/\alpha}\,\Omega$ is invertible on $\ell^2(\mathbb{Z}^2)$, the minimizing dual is given by the closed formula $\tilde g_\Lambda=\alpha\beta\,\Lambda^{-2}T^*_{g;1/\beta,1/\alpha}\,\Omega(\Omega\,T_{g;1/\beta,1/\alpha}\Lambda^{-2}T^*_{g;1/\beta,1/\alpha}\,\Omega)^{-1}\Omega\,e_{0,0}$. Concrete sufficient conditions (for instance diagonal $\Omega$ with weights $\omega_n$, and $\Lambda$ a multiplication operator) and examples showing improved frequency concentration were given, but only in those cases. Later work has developed the surrounding landscape extensively: Ron and Shen (1997) treated Weyl–Heisenberg frames in $L^2(\mathbb{R}^d)$; Jakobsen and Lemvig (2016) extended the Wexler–Raz biorthogonality relations, the Walnut and Janssen representations (two classical series formulas for the Gabor frame operator), and the duality principle (frame properties of a lattice are equivalent to duality properties of its window on the reciprocal lattice) to co-compact Gabor systems on locally compact abelian groups (the general class of topological abelian groups, containing $\mathbb{R}^d$, $\mathbb{Z}^d$, and finite groups); Jakobsen and Luef (2021) and Austad, Jakobsen, and Luef (2020) recast Gabor duality through Rieffel's Heisenberg modules (distinguished module classes over twisted group $C^*$-algebras — algebras of bounded operators closed under adjoints and operator-norm limits, with a twist from the time-frequency commutation phase) and Morita equivalence (a precise algebraic notion of equivalence under which two such algebras have equivalent categories of modules), generalizing the von Neumann algebra/coupling-constant viewpoint; Matusiak (2019) extended the Wexler–Raz relations and the Fundamental Identity of Gabor Analysis (the identity relating frame operators on a lattice and its reciprocal) to model sets (irregular, almost-periodic point sets). On the optimization side, Li, Liu, and Mi (2013) studied sparse duals and duals of minimal time/frequency support via $\ell^1$-type minimization in finite dimensions; Perraudin, Holighaus, Søndergaard, and Balazs (2014) computed dual windows satisfying the Wexler–Raz equations by convex optimization and proximal splitting; Christensen (2006) and Kiselev, Minin, Novikov, and Ushakov (2024) constructed compactly supported or well-localized dual pairs in specific regimes; and Díaz Martín, Lammers, Miller, and Powell (2026) began the theory of $L^1$-minimal dual windows, handling the 'painless' (Daubechies–Grossmann–Meyer) regime — lattice-window configurations for which the Gabor frame operator is a multiplication operator and hence explicitly invertible — and Gabor frames with diagonally dominant frame operators. On the numerical side, Del Prete (1999) and Strohmer (2001) studied decay, computation, and finite-dimensional approximation of dual windows, and Balan, Casazza, Heil, and Landau (2006) developed the localization theory of frames (quantitative off-diagonal decay of the matrix of frame inner products) that yields quantitative truncation results for localized Gabor systems. None of these supplies the general theory that Section 7's construction leaves open.

Problem Statement

Develop a general theory of dual Gabor windows optimized for non-canonical (non-L2) criteria, extending the quadratic Omega-Lambda construction of Section 7 of Daubechies-Landau-Landau (1994) beyond its case-by-case sufficient conditions. For a Gabor frame $\mathcal{G}(g,\alpha,\beta)$ in $L^2(\mathbb{R})$, the requested components, which jointly serve the single objective of making optimized dual windows a predictable, closed theory rather than a collection of examples, are: (1) given an arbitrary positive self-adjoint regularizer $\Lambda$ (modeling, e.g., smoothness, concentration, or support penalties), characterize the admissible complementary operators $\Omega$ on $\ell^2(\mathbb{Z}^2)$ — beyond diagonal weights — and prove the sharpest possible two-sided inequalities of the form $A'\|\Omega^{-1}c\|^2\le\|\Lambda^{-1}T^*_{g;1/\beta,1/\alpha}c\|^2\le B'\|\Omega^{-1}c\|^2$, $c\in\ell^2$; (2) give necessary and sufficient conditions for the invertibility on $\ell^2(\mathbb{Z}^2)$ of the operator $\Omega\,T_{g;1/\beta,1/\alpha}\Lambda^{-2}T^*_{g;1/\beta,1/\alpha}\,\Omega$ appearing in the closed-form minimizer, thereby characterizing exactly when that formula defines the optimal dual; (3) extend the optimization beyond quadratic penalizers: characterize existence, uniqueness, and stability (with respect to perturbations of the window and of the lattice parameters $\alpha,\beta$) of dual windows minimizing general convex functionals — such as $L^p$-type norms or sparsity-promoting criteria — subject to the Wexler-Raz duality constraints; and (4) establish general truncation and error-control bounds that quantify, as a function of the lattice parameters and the decay of the relevant inner products, how many lattice indices must be retained to compute the (optimized or canonical) dual window to a prescribed accuracy, going beyond spectral-gap estimates in specific cases. The extension of the von Neumann algebra and coupling-constant analysis to higher-dimensional and non-Weyl-Heisenberg group-generated systems is excluded from this open core because the later literature (Ron and Shen 1997; Jakobsen and Lemvig 2016; Jakobsen and Luef 2021; Austad, Jakobsen, and Luef 2020) has substantially carried it out. The four components (1)–(4) are jointly required parts of this single objective. Components (1) and (2) are not independent: for fixed $g$, $\Lambda$, and $\Omega$, the two-sided bound in (1) holds exactly when the operator $\Omega\,T_{g;1/\beta,1/\alpha}\Lambda^{-2}T^*_{g;1/\beta,1/\alpha}\,\Omega$ in (2) is bounded and bounded below, equivalently invertible, so establishing (1) also settles (2) and they may be answered jointly.

References

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