Sparsity of Gabor representation of Schrödinger propagators
type: Item
instance of: scholarly article
title: Sparsity of Gabor representation of Schrödinger propagators
zbMATH Open document ID: 1171.35133
author: Elena Cordero
author: Fabio Nicola
author: Luigi Rodino
published in: Applied and Computational Harmonic Analysis
publication date: 2009-04-21T00:00:00Z
review text: The authors consider Fourier integral operators given by \[ Tf(x)= \int_{\mathbb{R}^d}e^{2\pi:\Phi(x,\eta)}\sigma(x,\eta)\widehat f(\eta)d\eta,\;x\in\mathbb{R}^d, \] where the symbol \(\sigma\in BC^\infty(\mathbb{R}^{2d}\) and the phase function \(\Phi\in{\mathcal C}^\infty(\mathbb{R}^{2d};\mathbb{R})\) fulfills the estimates \(|\partial^\alpha_z\Phi(x)|\leq C_\alpha\), \(|\alpha|\geq 2\), \(z=(x,\eta)\in\mathbb{R}^{2d}\), and the nondegeneracy condition \(|\det \partial^r_{x,\eta}\Phi(x,\eta)|\geq\delta>0\). They prove the continuity of \(T\) on some modulation spaces \(M^p_\mu\), show that Gabor frames provide optimally sparse representations of such operators and present a formula which expresses \(Tf\), \(f\in M^p\equiv M^p_1\) as a finite sum of Gabor atoms, modulo an error whose size is estimated in terms of its time-frequency concentration. Numerical examples for the Schrödinger case (the free particle and the harmonic oscillator) demonstrate the fast computation of these operators.
reviewed by: Viorel Iftimie
Mathematics Subject Classification ID: 35S30
Mathematics Subject Classification ID: 65T99
Mathematics Subject Classification ID: 47G30
zbMATH DE Number: 5545831
zbMATH Keywords: Fourier integral operators
zbMATH Keywords: modulation spaces
zbMATH Keywords: short-time Fourier transform
zbMATH Keywords: Gabor frames
zbMATH Keywords: Schrödinger equation
MaRDI profile type: Publication
full work available at URL: j.acha.2008.08.003
OpenAlex ID: W2069565717
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DOI: 10.1016/J.ACHA.2008.08.003
description: scientific article; zbMATH DE number 5545831