d’Alembert functional equations in distributions
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- by E. Y. Deeba and E. L. Koh PDF
- Proc. Amer. Math. Soc. 116 (1992), 157-164 Request permission
Abstract:
In this paper we shall develop a method to define and solve the D’Alembert functional equation in distributions. We shall also show that for regular distributions (i.e., locally integrable functions) the distributional solution reduces to the classical one.References
- J. Aczél, The state of the second part of Hilbert’s fifth problem, Bull. Amer. Math. Soc. (N.S.) 20 (1989), no. 2, 153–163. MR 981872, DOI 10.1090/S0273-0979-1989-15749-2
- J. Aczél and J. Dhombres, Functional equations in several variables, Encyclopedia of Mathematics and its Applications, vol. 31, Cambridge University Press, Cambridge, 1989. With applications to mathematics, information theory and to the natural and social sciences. MR 1004465, DOI 10.1017/CBO9781139086578
- J. Aczél, H. Haruki, M. A. McKiernan, and G. N. Sakovič, General and regular solutions of functional equations characterizing harmonic polynomials, Aequationes Math. 1 (1968), 37–53. MR 279471, DOI 10.1007/BF01817556
- John A. Baker, Regularity properties of functional equations, Aequationes Math. 6 (1971), 243–248. MR 293283, DOI 10.1007/BF01819759
- John A. Baker, Functional equations, tempered distributions and Fourier transforms, Trans. Amer. Math. Soc. 315 (1989), no. 1, 57–68. MR 979965, DOI 10.1090/S0002-9947-1989-0979965-5
- John A. Baker, Functional equations, distributions and approximate identities, Canad. J. Math. 42 (1990), no. 4, 696–708. MR 1074230, DOI 10.4153/CJM-1990-036-1 —, Difference operators, distribution and functional equations, preprint.
- E. Y. Deeba and E. L. Koh, The Pexider functional equations in distributions, Canad. J. Math. 42 (1990), no. 2, 304–314. MR 1051731, DOI 10.4153/CJM-1990-017-6
- I. Fenyő, On the general solution of a functional equation in the domain of distributions, Aequationes Math. 3 (1969), 236–246. MR 611691, DOI 10.1007/BF01817444 M. Fréchet, Pri la funkcia equacio $f\left ( {x + y} \right ) = f\left ( x \right ) + f\left ( y \right )$, Enseign. Math. 15 (1913), 390-393. L. Hörmander, The analysis of linear partial differential operators. I, Springer-Verlag, Berlin, 1983.
- E. L. Koh, The Cauchy functional equations in distributions, Proc. Amer. Math. Soc. 106 (1989), no. 3, 641–646. MR 942634, DOI 10.1090/S0002-9939-1989-0942634-7
- A. Járai, On regular solutions of functional equations, Aequationes Math. 30 (1986), no. 1, 21–54. MR 837038, DOI 10.1007/BF02189909
- M. Neagu, About the Pompeiu equation in distributions, Inst. Politehn. Traian Vuia Timişoara Lucrăr. Sem. Mat. Fiz. May (1984), 62–66 (English, with Romanian summary). MR 783941
- Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, IX-X, Hermann, Paris, 1966 (French). Nouvelle édition, entiérement corrigée, refondue et augmentée. MR 0209834
- Halina Światak, On the regularity of the distributional and continuous solutions of the functional equations \[ \sum _{i=1}^{k}a_{l}(x,t)f(x+\varphi _{l}(t))= b(x,t)\], Aequationes Math. 1 (1968), 6–19. MR 279474, DOI 10.1007/BF01817554
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 116 (1992), 157-164
- MSC: Primary 46F10; Secondary 35L05, 39B52
- DOI: https://doi.org/10.1090/S0002-9939-1992-1100648-0
- MathSciNet review: 1100648