Two generalizations of Titchmarsh’s convolution theorem
HTML articles powered by AMS MathViewer
- by Raouf Doss
- Proc. Amer. Math. Soc. 108 (1990), 893-897
- DOI: https://doi.org/10.1090/S0002-9939-1990-1004416-8
- PDF | Request permission
Abstract:
Titchmarsh’s convolution theorem states that if the functions $f,g$ vanish on $( - \infty ,0)$ and if the convolution $f * g(t) = 0$ on an interval $(0,T)$, then there are two numbers $\alpha ,\beta \geq 0$ such that $\alpha + \beta = T,f = 0$ a.e. on $(0,\alpha )$, and $g = 0$ a.e. on $(0,\beta )$. $T$ may be infinite. For the case $T = \infty$ we prove that if $f * g = 0$ on $R$ and one of the two functions $f,g$ is 0 on $( - \infty ,0)$, then either $f$ or $g$ is 0 a.e. on $R$. Next we consider the integro-differential-difference equation $f * g(t) + \sum {{\lambda _{p\sigma }}{f^{(p)}}(t - {a_{p\sigma }}) = 0}$ for $t$ in $(0,T)$, where ${a_{\rho \sigma }} \geq 0,{\lambda _{p\sigma }}$ are constants. Conclusions similar to Titchmarsh’s hold with the additional information that $\alpha \geq T - {a_{\rho \sigma }}$ whenever ${\lambda _{\rho \sigma }} \ne 0$.References
- Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627
- M. M. Crum, On the resultant of two functions, Quart. J. Math. Oxford Ser. 12 (1941), 108–111. MR 4650, DOI 10.1093/qmath/os-12.1.108
- Raouf Doss, An elementary proof of Titchmarsh’s convolution theorem, Proc. Amer. Math. Soc. 104 (1988), no. 1, 181–184. MR 958063, DOI 10.1090/S0002-9939-1988-0958063-5
- Jacques Dufresnoy, Sur le produit de composition de deux fonctions, C. R. Acad. Sci. Paris 225 (1947), 857–859 (French). MR 22631
- Henry Helson, Harmonic analysis, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. MR 729682
- Kenneth Hoffman, Analytic functions and logmodular Banach algebras, Acta Math. 108 (1962), 271–317. MR 149330, DOI 10.1007/BF02545769
- Paul Koosis, On functions which are mean periodic on a half-line, Comm. Pure Appl. Math. 10 (1957), 133–149. MR 89297, DOI 10.1002/cpa.3160100106
- Peter D. Lax, Translation invariant spaces, Acta Math. 101 (1959), 163–178. MR 105620, DOI 10.1007/BF02559553
- Jacques-Louis Lions, Supports de produits de composition. I, C. R. Acad. Sci. Paris 232 (1951), 1530–1532 (French). MR 43254 J. Mikusinski, The Bochner integral, Academic Press, New York, 1978. E. C. Titchmarsh, The zeros of certain integral functions, Proc. London Math. Soc. 25 (1926), 283-302.
Bibliographic Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 893-897
- MSC: Primary 42A85; Secondary 45E10, 45J05
- DOI: https://doi.org/10.1090/S0002-9939-1990-1004416-8
- MathSciNet review: 1004416