[Paper Review] The Gabor wave front set

Luigi Rodino, Patrik Wahlberg|arXiv (Cornell University)|Jul 24, 2012

Mathematical Analysis and Transform Methods22 references4 citations

TL;DR

This paper introduces the Gabor wave front set, a new microlocal analysis tool defined via rapid decay of Gabor coefficients in conic subsets of phase space. It proves that the Gabor wave front set coincides with Hörmander's global wave front set and establishes invariance under phase-space shifts and inclusion under Weyl quantization of order-zero symbols, providing a time-frequency framework for singularities in PDEs.

ABSTRACT

We define the Gabor wave front set $WF_G(u)$ of a tempered distribution $u$ in terms of rapid decay of its Gabor coefficients in a conic subset of the phase space. We show the inclusion $$WF_G(a^w(x,D) u) \subseteq WF_G(u), u \in \mathscr S'(\mathbb R^d), a \in S_{0,0}^0,$$ where $S_{0,0}^0$ denotes the Hörmander symbol class of order zero and parameter values zero. We compare our definition with other definitions in the literature, namely the classical and the global wave front sets of Hörmander, and the $\cS$-wave front set of Coriasco and Maniccia. In particular, we prove that the Gabor wave front set and the global wave front set of Hörmander coincide.

Motivation & Objective

• To define a new microlocal singularity set, the Gabor wave front set, based on time-frequency analysis.
• To establish its invariance under phase-space translations and modulations.
• To prove that the Gabor wave front set coincides with Hörmander's global wave front set.
• To show that pseudodifferential operators with symbols in $ S_{0,0}^0 $ preserve the Gabor wave front set.
• To compare the Gabor wave front set with other wave front sets, including the $ \mathscr{S} $-wave front set of Coriasco and Maniccia.
• To demonstrate that the Gabor wave front set captures finer singularities than classical or $ \mathscr{S} $-wave front sets in certain cases.

Proposed method

• Define the Gabor wave front set $ WF_G(u) $ via rapid decay of Gabor coefficients $ c_λ = (u, \Pi(\lambda)\varphi) $ over a lattice $ \Lambda \subset \mathbb{R}^{2d} $, for $ \lambda \in \Lambda \cap \Gamma $, where $ \Gamma $ is a conic neighborhood.
• Use the short-time Fourier transform (STFT) $ V_\varphi u(x,\xi) = (u, M_\xi T_x \varphi) $ to define a continuous version of the Gabor wave front set.
• Prove that the Gabor wave front set is invariant under phase-space shifts $ \Pi(z)u $, i.e., $ WF_G(\Pi(z)u) = WF_G(u) $.
• Establish the inclusion $ WF_G(a^w(x,D)u) \subseteq WF_G(u) $ for $ a \in S_{0,0}^0 $, using estimates on the decay of Gabor coefficients.
• Show that the Gabor wave front set is independent of the choice of non-zero Schwartz window $ \varphi $, provided the Gabor system forms a frame.
• Prove that $ WF_G(u) = WF_{\text{global}}(u) $, the global wave front set of Hörmander, by relating decay conditions in the lattice setting to the continuous STFT.

Experimental results

Research questions

• RQ1 Can a wave front set be defined intrinsically using Gabor frames and the short-time Fourier transform?
• RQ2 Does the Gabor wave front set coincide with Hörmander's global wave front set?
• RQ3 Is the Gabor wave front set invariant under phase-space translations and modulations?
• RQ4 Does the Gabor wave front set satisfy the inclusion $ WF_G(a^w(x,D)u) \subseteq WF_G(u) $ for $ a \in S_{0,0}^0 $?
• RQ5 How does the Gabor wave front set compare with the $ \mathscr{S} $-wave front set of Coriasco and Maniccia in terms of singularity detection?

Key findings

• The Gabor wave front set $ WF_G(u) $ is defined as the set of points $ z_0 \in \mathbb{R}^{2d} \setminus \{0\} $ such that $ \sup_{\lambda \in \Lambda \cap \Gamma} \langle \lambda \rangle^N |c_\lambda| < \infty $ for all $ N \geq 0 $, where $ \Gamma $ is a conic neighborhood of $ z_0 $.
• The Gabor wave front set is invariant under phase-space shifts: $ WF_G(\Pi(z)u) = WF_G(u) $ for all $ z \in \mathbb{R}^{2d} $.
• The Gabor wave front set coincides exactly with Hörmander's global wave front set: $ WF_G(u) = WF_{\text{global}}(u) $.
• For $ a \in S_{0,0}^0 $, the inclusion $ WF_G(a^w(x,D)u) \subseteq WF_G(u) $ holds, and this is strengthened to $ WF_G(a^w(x,D)u) \subseteq WF_G(u) \cap \operatorname{conesupp}(a) $ in the final section.
• For the Dirac delta $ \delta_{x_0} $, $ WF_G(\delta_{x_0}) = \{0\} \times (\mathbb{R}^d \setminus \{0\}) $, matching the classical wave front set but differing from the $ \mathscr{S} $-wave front set.
• For the plane wave $ u(x) = e^{i\langle x,\xi_0\rangle} $, $ WF_G(u) = (\mathbb{R}^d \setminus \{0\}) \times \{0\} $, showing that the Gabor wave front set detects oscillatory singularities not captured by classical wave front sets.