by Joonas Ilmavirta, Keijo Mönkkönen
Abstract:
We prove that if~$P(D)$ is some constant coefficient partial differential operator and~$f$ is a scalar field such that~$P(D)f$ vanishes in a given open set, then the integrals of~$f$ over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.
Reference:
Partial data problems in scalar and vector field tomography (Joonas Ilmavirta, Keijo Mönkkönen), Journal of Fourier Analysis and Applications, volume 28, number 34, 2022.
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We prove that if $P(D)$ is some constant coefficient partial differential operator and $f$ is a scalar field such that $P(D)f$ vanishes in a given open set, then the integrals of $f$ over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.
[arXiv]
Bibtex Entry:
@article{polynomial-ucp,
author = {Joonas Ilmavirta and Keijo M{\"o}nkk{\"o}nen},
title = {{Partial data problems in scalar and vector field tomography}},
year = 2022,
month = mar,
journal = {Journal of Fourier Analysis and Applications},
volume = 28,
number = 34,
arxiv = {2103.14385},
url={http://users.jyu.fi/~jojapeil/pub/polynomial-ucp.pdf},
abstract = {We prove that if~$P(D)$ is some constant coefficient partial differential operator and~$f$ is a scalar field such that~$P(D)f$ vanishes in a given open set, then the integrals of~$f$ over all lines intersecting that open set determine the scalar field uniquely everywhere. This is done by proving a unique continuation property of fractional Laplacians which implies uniqueness for the partial data problem. We also apply our results to partial data problems of vector fields.},
doi = {10.1007/s00041-022-09907-9}
}