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Characterization of eigenfunctions of the Laplacian by boundedness conditions


Author: Robert S. Strichartz
Journal: Trans. Amer. Math. Soc. 338 (1993), 971-979
MSC: Primary 42B10; Secondary 35J05, 35P05, 43A80
DOI: https://doi.org/10.1090/S0002-9947-1993-1108614-1
MathSciNet review: 1108614
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Abstract: If $ {\{ {f_k}(x)\} _{k \in \mathbb{Z}}}$ is a doubly infinite sequence of functions on $ {\mathbb{R}^n}$ which are uniformly bounded and such that $ \Delta {f_k} = {f_{k + 1}}$, then $ \Delta {f_0} = - {f_0}$. This generalizes a theorem of Roe $ (n = 1)$. The analogous statement is true on the Heisenberg group, but false in hyperbolic space.


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  • [Fo] G. Follard, Harmonic analysis in phase space, Ann. of Math. Stud., no. 122, Princeton Univ. Press, Princeton, N. J., 1989. MR 983366 (92k:22017)
  • [Ga 1] J.-P. Gabardo, Tempered distributions with spectral gaps, Math. Proc. Cambridge Philos. Soc. 106 (1989), 143-162. MR 994086 (90k:42027)
  • [Ga 2] -, Tempered distributions supported on a half-space of $ {\mathbb{R}^n}$ and their Fourier transforms, Canad. J. Math. 43 (1991), 61-88. MR 1108914 (92h:46056)
  • [Ge 1] D. Geller, Fourier analysis on the Heisenberg group. I: Schwartz space, J. Funct. Anal. 36 (1980), 205-254. MR 569254 (81g:43008)
  • [Ge 2] -, Liouville's theorem for homogeneous groups, Comm. Partial Differential Equations 8 (1983), 1665-1677. MR 729197 (85f:58109)
  • [GS] I. M. Gelfand and G. E. Shilov, Generalized functions, vol. I, Academic Press, New York, 1964. MR 0166596 (29:3869)
  • [He] S. Helgason, Groups and geometric analysis, Academic Press, New York, 1984. MR 754767 (86c:22017)
  • [Ho] R. Howard, A note on Roe's characterization of the sine function, Proc. Amer. Math. Soc. 105 (1989), 658-663. MR 942633 (89g:33001)
  • [HR] R. Howard and M. Reese, Characterization of eigenfunctions by boundedness conditions, Canad. Math. Bull. 35 (1992), 204-213. MR 1165169 (93e:35077)
  • [R] J. Roe, A characterization of the sine function, Math. Proc. Cambridge Philos. Soc. 87 (1980), 69-73. MR 549299 (81a:33001)
  • [RS] M. Reed and B. Simon, Methods of mathematical physics. Vol. II. Functional analysis, Academic Press, New York, 1980. MR 751959 (85e:46002)
  • [SW] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N. J., 1971. MR 0304972 (46:4102)
  • [Str 1] R. Strichartz, Harmonic analysis as spectral theory of Laplacians, J. Funct. Anal. 87 (1989), 51-148; Corrigendum, 109 (1992), 457-460. MR 1025883 (91c:43015)
  • [Str 2] -, $ {L^p}$ harmonic analysis and Radon transforms on the Heisenberg group, J. Funct. Anal. 96 (1991), 350-406. MR 1101262 (92d:22015)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1108614-1
Article copyright: © Copyright 1993 American Mathematical Society

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