Microlocal analaysis of the Doppler transform on \(\mathbb{R}^3\)
type: Item
instance of: scholarly article
title: Microlocal analaysis of the Doppler transform on \(\mathbb{R}^3\)
zbMATH Open document ID: 1064.35223
author: Karthik Ramaseshan
published in: The Journal of Fourier Analysis and Applications
publication date: 2004-10-01T00:00:00Z
review text: The author considers the Doppler transform of a vector field \(F= (f_1,f_2,f_3)\) on \(\mathbb{R}^3\), defined by \[ DF(x, p)= \sum_{j= 1,2,3} \int_{\mathbb{R}} p_jf_j(x+ tp)\,dt, \] where \(x\in\mathbb{R}^3\) and \(p\in\mathbb{R}^3\) with \(|p|= 1\). The problem under consideration is the reconstruction of curl \(F\) from \(DF\), known only for a small subset of lines in \(\mathbb{R}^3\). The author gives an answer in terms of microlocal techniques, by finding a parametrix of the problem, in the case of \(DF\)-restricted to all lines passing through a fixed smooth curve. Namely, \(D\) is regarded in the frame of the Fourier integral operators with singular symbols introduced by \textit{V . Guillemin} and \textit{G . Uhlmann} [Duke Math. J. 48, 251--267 (1981; Zbl 0462.58030)]. The corresponding symbol is proved to be elliptic (modulo the symbol of the curl) and the parametrix is then constructed by symbolic calculus.
reviewed by: Luigi Rodino
Mathematics Subject Classification ID: 35S30
Mathematics Subject Classification ID: 92C55
zbMATH DE Number: 2105067
zbMATH Keywords: parametrix
zbMATH Keywords: Fourier integral operators with singular symbols
zbMATH Keywords: symbolic calculus
MaRDI profile type: Publication
full work available at URL: s00041-004-8004-4
OpenAlex ID: W2049024326
DOI: 10.1007/S00041-004-8004-4
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description: scientific article; zbMATH DE number 2105067