Unique Determination of the Shape of a Scattering Screen from a Passive Measurement
Abstract
:1. Introduction
1.1. Antennas
1.2. Mathematical Background
1.3. Definitions and Theorems
- represent variables in , and we associate to them various projections described below.
- mean variables in or projections to . For example if then in that context , but we could have in an integral over a subset of without having to define the variable y separately.
- denote lifts to , meaning . For example if then . This notation can also be used as a projection . So, if then . Essentially and but we do not use this combined notation explicitly.
- is reserved for the fundamental solution to , defined in Lemma 2.
- mean the function u restricted to and , respectively. If their variable is in then they are the two-sided limits (traces) as . We often use and . These are simply the derivatives in the -direction of and , respectively. Often this is evaluated on where it then denotes the one-sided derivative, i.e., the trace of .
- : this is the set of distributions whose support is contained in , where we recall that signifies the shape of a screen .
2. Representation Theorems
- for all ,
- , and
- .
3. Solving the Inverse Problem
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Blåsten, E.; Päivärinta, L.; Sadique, S. Unique Determination of the Shape of a Scattering Screen from a Passive Measurement. Mathematics 2020, 8, 1156. https://doi.org/10.3390/math8071156
Blåsten E, Päivärinta L, Sadique S. Unique Determination of the Shape of a Scattering Screen from a Passive Measurement. Mathematics. 2020; 8(7):1156. https://doi.org/10.3390/math8071156
Chicago/Turabian StyleBlåsten, Emilia, Lassi Päivärinta, and Sadia Sadique. 2020. "Unique Determination of the Shape of a Scattering Screen from a Passive Measurement" Mathematics 8, no. 7: 1156. https://doi.org/10.3390/math8071156
APA StyleBlåsten, E., Päivärinta, L., & Sadique, S. (2020). Unique Determination of the Shape of a Scattering Screen from a Passive Measurement. Mathematics, 8(7), 1156. https://doi.org/10.3390/math8071156