Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Wave equations with finite velocity of propagation

Author: Stephen J. Berman
Journal: Trans. Amer. Math. Soc. 188 (1974), 127-148
MSC: Primary 35L05; Secondary 43A70
MathSciNet review: 0330781
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: If B is a selfadjoint translation-invariant operator on the space $ {L^2}$ of complex-valued functions on n-dimensional Euclidean space which are square-summable with respect to Lebesgue measure, then the wave equation $ {d^2}F/d{t^2} + {B^2}F = 0$ has the solution $ F(t) = (\cos tB)f + ((\sin tB)/B)g$, for f and g in $ {L^2}$. In the classical case in which $ - {B^2}$ is the Laplacian, this solution has finite velocity of propagation in the sense that (letting supp denote support of a function) $ {\text{supp}}\;F(t) \subset ({\text{supp}}\;f \cup {\text{supp}}\;g) + {K_t}$ for all f and g and some compact set $ {K_t}$ independent of f and g. We show that a converse holds, namely, if $ \cos \;tB$ has finite velocity of propagation (that is, if $ {\text{supp}}\,(\cos tB)f \subset {\text{supp}}\;f + {K_t}$ for all f and some compact $ {K_t}$) for three values of t whose reciprocals are independent over the rationals, then $ {B^2}$ must be a second order differential operator.

If Euclidean space is replaced by a locally compact abelian group which does not contain the real line as a subgroup, then $ \cos \;tB$ has finite velocity of propagation for all t if and only if it is convolution with a distribution $ {T_t}$ such that all $ {T_t}$ are supported on a compact open subgroup.

Problems of a similar nature are discussed for compact connected abelian groups and for the nonabelian group $ {\text{SL}}(2,{\mathbf{R}})$.

References [Enhancements On Off] (What's this?)

  • [1] Stephen Berman, Wave equations with finite velocity of propagation, Ph. D. Thesis, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass., 1972.
  • [2] Francois Bruhat, Distributions sur une groupe localement compact et applications à l'étude des représentations des groupes p-adiques, Bull. Math. Soc. France 89 (1961), 43-75. MR 25 #4354. MR 0140941 (25:4354)
  • [3] L. Ehrenpreis and F. I. Mautner, Some properties of the Fourier transform on semisimple Lie groups. I, Ann. of Math. (2) 61 (1955), 406-439. MR 16, 1017. MR 0069311 (16:1017b)
  • 1. -, Some properties of the Fourier transform on semisimple Lie groups. II, Trans. Amer. Math. Soc. 84 (1957), 1-55. MR 18, 745. MR 0083683 (18:745f)
  • [4] Y. Fourès Bruhat and I. E. Segal, Causality and analyticity, Trans. Amer. Math. Soc. 78 (1955), 385-405. MR 16, 1032. MR 0069401 (16:1032d)
  • [5] Robert Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1965. MR 31 #4927. MR 0180696 (31:4927)
  • [6] E. Hewitt and K. Ross, Abstract harmonic analysis. Vol. I, Structure of topological groups. Integration theory, group representations, Die Grundlehren der math. Wissenschaften, Band 115, Academic Press, New York; Springer-Verlag, Berlin, 1963. MR 18, 158. MR 551496 (81k:43001)
  • [7] J.-L. Lions, Support dans la transformation de Laplace, J. Analyse Math. 2 (1953), 369-380. MR 15, 307. MR 0058013 (15:307g)
  • [8] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 271-355. MR 19, 280. MR 0086990 (19:280a)
  • [9] I. G. Petrovskiĭ, Partial differential equations, Fizmatgiz, Moscow, 1961; English transl., Scripta Technica; distributed by Saunders, Philadelphia, Pa., 1967. MR 25 #2308; MR 35 #1906. MR 0211021 (35:1906)
  • [10] Walter Rudin, Fourier analysis on groups, Interscience, New York and London, 1962. MR 0152834 (27:2808)
  • [11] S. Saks and A. Zygmund, Analytic functions, Monografie Mat., Tom 10, PWN, Warsaw, 1938; English transl., Monografie Mat., Tom 28, PWN, Warsaw, 1952. MR 14, 1073. MR 0055432 (14:1073a)
  • [12] L. Schwartz, Théorie des distributions. Tomes I, II, Actualités Sci. Indust., nos. 1901, 1122, Hermann, Paris, 1950, 1951. MR 12, 31; MR 12, 833.
  • [13] I. E. Segal, A non-commutative extension of abstract integration, Ann. of Math. (2) 57 (1953), 401-457. MR 14, 991. MR 0054864 (14:991f)
  • [14] E. C. Titchmarsh, The theory of functions, 2nd ed., Oxford Univ. Press, Oxford, 1939.
  • [15] Andre Weil, L'intégration dans les groupes topologiques et ses applications, 2nd ed., Hermann, Paris, 1951.
  • [16] A. Zygmund, Trigonometrical series. Vols. 1, 2, 2nd rev. ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498. MR 0107776 (21:6498)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35L05, 43A70

Retrieve articles in all journals with MSC: 35L05, 43A70

Additional Information

Keywords: Wave equation, locally compact abelian group, translation-invariant operators, Laplacian, velocity of propagation, distributions on a LCA group
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society