×

A note on directional wavelet transform and the propagation of polarization sets for solutions of Maxwell’s equations. (English) Zbl 1302.42052

The authors represent the solutions of Maxwell’s equations in terms of the directional wavelet transform. Then, they make use of the polarization set, which refines the notion of wavefront set for vector-valued distributions as introduced by N. Dencker [Ark. Mat. 20, 23–60 (1982; Zbl 0503.58031)], in order to analyze the singularities of the solutions of Maxwell’s equations.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
35A18 Wave front sets in context of PDEs
58K45 Singularities of vector fields, topological aspects
35A21 Singularity in context of PDEs
35Q61 Maxwell equations

Citations:

Zbl 0503.58031
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] F. A. Apolonio, D. H. T. Franco and F. N. Fagundes, Int. J. Math. Math. Sc. 2012, 758694 (2012), DOI: 10.1155/2012/758694.
[2] N. Dencker, J. Funct. Anal. 46, 351 (1982), DOI: 10.1016/0022-1236(82)90051-9. genRefLink(128, ’rf2’, ’A1982NX42100003’);
[3] M. Kline, Electromagnetic theory and geometrical optics, Research Report No. EM-171, Courant Institute of Mathematical Research (1962) . · Zbl 0123.23602
[4] G. Kaiser, Phys. Lett. A 168, 28 (1992), DOI: 10.1016/0375-9601(92)90324-F. genRefLink(128, ’rf4’, ’A1992JH67100006’); genRefLink(64, ’rf4’, ’1992PhLA..168...28K’);
[5] G.   Kaiser , A Friendly Guide to Wavelets ( Birkhäuser , 1999 ) . · Zbl 1230.42001
[6] L.   Debnath and P.   Mikusiński , Introduction to Hilbert Spaces , 3rd edn. ( Elsevier Academic Press , 2005 ) . · Zbl 0715.46009
[7] J.-P.   Antoine , Two-dimensional Wavelets and Their Relatives ( Cambridge University Press , 2004 ) . genRefLink(16, ’rf7’, ’10.1017
[8] R. S. Pathak, Tohoku Math. J. 56, 411 (2004), DOI: 10.2748/tmj/1113246676. genRefLink(128, ’rf8’, ’000225235600009’);
[9] L. Hörmander, Comm. Pure Appl. Math. 24, 671 (1971). genRefLink(128, ’rf9’, ’A1971L010800004’);
[10] K. Nishiwada, Surikaiseki-kenkyusho Kokyuroku, RIMS, Kyoto Univ. 239, 19 (1975).
[11] K. Nishiwada, Publ. RIMS, Kyoto Univ. 14, 309 (1978), DOI: 10.2977/prims/1195189065. genRefLink(128, ’rf11’, ’000317607900006’);
[12] L.   Schwartz , Méthodes Mathématiques Pour les Sciences Physiques , 2nd edn. ( Hermann , 1979 ) .
[13] L. Hörmander, Acta Math. 127, 79 (1971). genRefLink(128, ’rf13’, ’A1971J664400005’);
[14] L. Schwartz, Lectures on partial differential equations and representations of semi-groups, Notes by K. Varadarajan, Tata Institute of Fundamental Research, Bombay (1957) .
[15] J.-N. Wang, Inv. Probl. 14, 733 (1998), DOI: 10.1088/0266-5611/14/3/021. genRefLink(128, ’rf15’, ’000074497500021’); genRefLink(64, ’rf15’, ’1998InvPr..14..733W’);
[16] J. J. Duistermaat and L. Hörmander, Acta Math. 128, 183 (1972), DOI: 10.1007/BF02392165. genRefLink(128, ’rf16’, ’000207939800009’);
[17] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Westview Press, Reprint edition 1995) .
[18] M.   Cheney and B.   Borden , Fundamentals of Radar Imaging ( Society for Industrial and Applied Mathematics , Philadelphia , 2009 ) . genRefLink(16, ’rf18’, ’10.1137 · Zbl 1192.78002
[19] D. H. T. Franco, On the relation between singularities of incident and reflected waves to synthetic aperture radar data, work in progress .
[20] E. J. Candes and D. L. Donoho, Appl. Comp. Harm. Anal. 19, 162 (2003). genRefLink(128, ’rf20’, ’000231764700002’);
[21] G. Kutyniok and D. Labate, Trans. Amer. Math. Soc. 361, 2719 (2009), DOI: 10.1090/S0002-9947-08-04700-4. genRefLink(128, ’rf21’, ’000263773800022’);
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.