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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
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] Louis Auslander, Lecture notes on nil-theta functions, American Mathematical Society, Providence, R.I., 1977. Regional Conference Series in Mathematics, No. 34. MR 0466409
  • [2] Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917
  • [3] Roger Howe, Quantum mechanics and partial differential equations, J. Funct. Anal. 38 (1980), no. 2, 188–254. MR 587908,
  • [4] David Mumford, Tata lectures on theta. I, Progress in Mathematics, vol. 28, Birkhäuser Boston, Inc., Boston, MA, 1983. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. MR 688651
  • [5] W. Schempp, Gruppentheoretische Aspekte der Signalübertragung und der kardinalen Interpolationssplines. I, Math. Methods Appl. Sci. 5 (1983), no. 2, 195–215 (German, with English summary). MR 703955,
  • [6] Walter Schempp, Radar ambiguity functions, nilpotent harmonic analysis, and holomorphic theta series, Special functions: group theoretical aspects and applications, Math. Appl., Reidel, Dordrecht, 1984, pp. 217–260. MR 774060
  • [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, Second edition, Pergamon Press, Oxford-Edinburgh-New York-Paris-Frankfurt, 1964. MR 0180402

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Article copyright: © Copyright 1984 American Mathematical Society

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