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Radar ambiguity functions, the Heisenberg group, and holomorphic theta series


Author: Walter Schempp
Journal: Proc. Amer. Math. Soc. 92 (1984), 103-110
MSC: Primary 22E25; Secondary 22E30, 33A75, 43A80, 60G35, 81D05, 94A12
DOI: https://doi.org/10.1090/S0002-9939-1984-0749901-6
MathSciNet review: 749901
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Abstract: The concept of linear Schrödinger representation of the real Heisenberg nilpotent group and its various realizations is used to link the theory of radar ambiguity functions with nilpotent harmonic analysis. This group-representation theoretic approach allows us to analyze the radar ambiguity functions simultaneously in time and frequency. Moreover, it allows us to determine the group of all transformations that leave the radar ambiguity surfaces invariant and to specify all admissible envelope functions that belong to radar signals of the same finite energy. In particular, an investigation of the radial, i.e., S0(2, R)-invariant radar ambiguity surfaces, gives rise to an identity for Laguerre-Weber functions of different orders, which implies on its part an identity for holomorphic theta series.


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  • [1] L. Auslander, Lecture notes on nil-theta functions, CBMS Regional Conf. Ser. in Math., no. 34, Amer. Math. Soc., Providence, R.I., 1977. MR 0466409 (57:6289)
  • [2] A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Ann. of Math. Studies, no. 94, Princeton Univ. Press, Princeton, N.J., 1980. MR 554917 (83c:22018)
  • [3] R. Howe, Quantum mechanics and partial differential equations, J. Funct. Anal. 38 (1980), 188-254. MR 587908 (83b:35166)
  • [4] D. Mumford, Tata lectures on theta. I, Progress in Math., Vol. 28, Birkhäuser, Boston, Mass., 1983. MR 688651 (85h:14026)
  • [5] W. Schempp, Gruppentheoretische Aspekte der Signalübertragung und der kardinalen Interpolations-splines. I, Math. Methods Appl. Sci. 5 (1983), 195-215. MR 703955 (85c:22011a)
  • [6] -, Radar ambiguity functions, nilpotent harmonic analysis, and holomorphic theta series, Special Functions. Group Theoretic Aspects and Applications (R. A. Askey, T. H. Koornwinder, W. Schempp, eds.), Reidel, Dordrecht, 1984. MR 774060 (86k:22021)
  • [7] -, Harmonic analysis on the Heisenberg group with applications, Pitman, Boston, Mass.
  • [8] -, The complex Laplace-Beltrami operator, nilpotent harmonic analysis, and holomorphic theta series (to appear).
  • [9] C. H. Wilcox, The synthesis problem for radar ambiguity functions, MRC Tech. Summary Report #157, 1960.
  • [10] P. M. Woodward, Probability and information theory with applications to radar, 2nd ed., Pergamon Press, New York, 1964. MR 0180402 (31:4637)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1984-0749901-6
Article copyright: © Copyright 1984 American Mathematical Society

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