Harmonic Analysis/Mathematical Analysis
On the Lévy–Raikov–Marcinkiewicz theorem
[Sur le théorème de Lévy–Raikov–Marcinkiewicz]

Présenté par : Jean-Pierre Kahane

Ostrovskii, Iossif1, 2 ; Ulanovskii, Alexander3
1 Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey
2 Verkin Institute for Low Temperature Physics and Engineering, 61103 Kharkov, Ukraine
3 Stavanger University College, PO Box 2557, Ullandhaug, 4091 Stavanger, Norway
Comptes Rendus. Mathématique, Tome 336 (2003) no. 3, pp. 237-240.

Soit une mesure de Borel μ, finie et non-négative. Le théorème classique de Lévy–Raikov–Marcinkiewicz affirme que la transformée de Fourier μ ^ de μ a une continuation analytique dans la bande {t:0<ℑt<R} si elle a une continuation analytique dans quelque demi-voisinage complexe de l'origine contenant un intervalle (0,iR). Nous prolongeons ce résultat à des classes générales de mesures et de distributions en supposant la non-négativité sur un rayon et une croissance tempérée sur toute la ligne.

Let μ be a finite nonnegative Borel measure. The classical Lévy–Raikov–Marcinkiewicz theorem states that if its Fourier transform μ ^ can be analytically continued to some complex half-neighborhood of the origin containing an interval (0,iR) then μ ^ admits analytic continuation into the strip {t:0<ℑt<R}. We extend this result to general classes of measures and distributions, assuming non-negativity only on some ray and allowing temperate growth on the whole line.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00035-9
Ostrovskii, Iossif 1, 2 ; Ulanovskii, Alexander 3

1 Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey
2 Verkin Institute for Low Temperature Physics and Engineering, 61103 Kharkov, Ukraine
3 Stavanger University College, PO Box 2557, Ullandhaug, 4091 Stavanger, Norway 
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     title = {On the {L\'evy{\textendash}Raikov{\textendash}Marcinkiewicz} theorem},
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     pages = {237--240},
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Ostrovskii, Iossif; Ulanovskii, Alexander. On the Lévy–Raikov–Marcinkiewicz theorem. Comptes Rendus. Mathématique, Tome 336 (2003) no. 3, pp. 237-240. doi : 10.1016/S1631-073X(03)00035-9. http://www.numdam.org/articles/10.1016/S1631-073X(03)00035-9/

[1] Hörmander, L. The Analysis of Linear Partial Differential Operators, I, Springer-Verlag, Berlin, 1983

[2] Linnik, Ju.V.; Ostrovskii, I.V. Decomposition of Random Variables and Vectors, American Mathematical Society, Providence, RI, 1977

[3] Ostrovskii, I.V.; Ulanovskii, A. On sign changes of distributions having spectral gap at the origin, C. R. Acad. Sci. Paris, Sér. I, Volume 336 (2003) (to be published)

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