Mathematical Analysis
A characterization of the Fourier transform and related topics
Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 625-628.

It is shown that the Fourier transform is essentially, up to a simple adjustment, the only transform on the corresponding space which maps convolutions to products and products to convolutions (surprisingly, no linearity is assumed a priori).

On montre que la transformation de Fourier est essentiellement, à une simple adaptation près, la seule application, qui sur les espaces où elle opère, transforme les convolutions en produits et les produits en convolutions. (De manière surprenante la linéarité n'est pas supposée à priori.)

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.04.001
Alesker, Semyon 1; Artstein-Avidan, Shiri 1; Milman, Vitali 1

1 Department of Mathematics, Tel Aviv University, 69978 Tel Aviv, Israel
@article{CRMATH_2008__346_11-12_625_0,
     author = {Alesker, Semyon and Artstein-Avidan, Shiri and Milman, Vitali},
     title = {A characterization of the {Fourier} transform and related topics},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {625--628},
     publisher = {Elsevier},
     volume = {346},
     number = {11-12},
     year = {2008},
     doi = {10.1016/j.crma.2008.04.001},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2008.04.001/}
}
TY  - JOUR
AU  - Alesker, Semyon
AU  - Artstein-Avidan, Shiri
AU  - Milman, Vitali
TI  - A characterization of the Fourier transform and related topics
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 625
EP  - 628
VL  - 346
IS  - 11-12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2008.04.001/
DO  - 10.1016/j.crma.2008.04.001
LA  - en
ID  - CRMATH_2008__346_11-12_625_0
ER  - 
%0 Journal Article
%A Alesker, Semyon
%A Artstein-Avidan, Shiri
%A Milman, Vitali
%T A characterization of the Fourier transform and related topics
%J Comptes Rendus. Mathématique
%D 2008
%P 625-628
%V 346
%N 11-12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.04.001/
%R 10.1016/j.crma.2008.04.001
%G en
%F CRMATH_2008__346_11-12_625_0
Alesker, Semyon; Artstein-Avidan, Shiri; Milman, Vitali. A characterization of the Fourier transform and related topics. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 625-628. doi : 10.1016/j.crma.2008.04.001. http://www.numdam.org/articles/10.1016/j.crma.2008.04.001/

[1] S. Artstein-Avidan, V. Milman, The Concept of Duality in Convex Analysis, and the Characterization of the Legendre Transform. Annals of Mathematics, in press

[2] S. Artstein-Avidan, V. Milman, A characterization of the concept of duality, in: Electronic Research Announcements in Mathematical Sciences, AIMS, vol. 14, 2007, pp. 48–65

[3] Gel'fand, I.M.; Shilov, G.E. Generalized Functions. Vol. I: Properties and Operations, Academic Press, New York–London, 1964 (Translated by Eugene Saletan)

[4] Stein, E.M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993 (With the assistance of Timothy S. Murphy Monographs in Harmonic Analysis, III) (ISBN: 0-691-03216-5)

[5] Stein, E.M.; Shakarchi, R. Complex Analysis, Princeton Lectures in Analysis, vol. II, Princeton University Press, Princeton, NJ, 2003 (ISBN: 0-691-11385-8)

Cited by Sources:

The research was supported in part by Israel Science Foundation: first named author by grant No. 1369/04, second named author by grant No. 865/07, third named author by grant No. 491/04. The second and third names authors were supported in part by BSF grant No. 2006079.