Comptes Rendus
no. 4
Harmonic Analysis/Mathematical Analysis
On sign changes of tempered distributions having a spectral gap at the origin
[Sur les changements du signe de distributions tempérées ayant une lacune spectrale à l'origine]

Présenté par : Jean-Pierre Kahane

Iossif Ostrovskii1, 2 ; Alexander Ulanovskii3
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, Volume 336 (2003) no. 4, pp. 325-330.

Il est connu que si une mesure de Borel réelle a une lacune spectrale à l'origine, alors ou la mesure doit avoir beaucoup de changements du signe ou elle est zéro identiquement. Supposons que la transformée de Fourier d'une distribution tempérée réelle coïncide dans un voisinage de l'origine avec la somme d'une fonction analytique et d'une série trigonometrique lacunaire. Nous conjecturons que ou elle coïncide avec la somme sur toute la ligne réelle ou la distribution doit avoir beaucoup de changements du signe. La Note contient quelques résultats reliés à la conjecture. En particulier les résultats impliquent qu'une distribution tempérée réelle ayant une lacune spectrale à l'origine doit avoir beaucoup d'oscillations d'une grande amplitude.

It is known that if a real finite Borel measure has a spectral gap at the origin then either it must have many sign changes or it is zero identically. Assume the Fourier transform of a real temperate distribution agrees in a neighborhood of the origin with the sum of an analytic function and a lacunary trigonometric series. We conjecture that either the distribution must have many sign changes or the Fourier transform agrees with the sum on the whole line. The Note contains some results related to the conjecture. In particular, our results imply that a real temperate measure having spectral gap at the origin must have many oscillations with large amplitudes.

Reçu le :
Accepté le :
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DOI : 10.1016/S1631-073X(03)00036-0
Iossif Ostrovskii 1, 2 ; Alexander Ulanovskii 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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Iossif Ostrovskii; Alexander Ulanovskii. On sign changes of tempered distributions having a spectral gap at the origin. Comptes Rendus. Mathématique, Volume 336 (2003) no. 4, pp. 325-330. doi : 10.1016/S1631-073X(03)00036-0. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00036-0/

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

[2] P. Koosis The Logarithmic Integral, Vol. 2, Cambridge University Press, 1992

[3] B.Ja. Levin Distribution of Zeros of Entire Functions, American Mathematical Society, Providence, RI, 1980

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

[5] B.F. Logan, Jr., Properties of high-pass signals, Thesis, Electrical Engineering Department, Columbia University, New York, 1965

[6] I.V. Ostrovskii, A. Ulanovskii, On the Lévy–Raikov–Marcinkiewicz theorem, C. R. Acad. Sci. Paris, submitted

[7] H.S. Shapiro Functions with spectral gap, Bull. Amer. Math. Soc., Volume 79 (1973) no. 2, pp. 354-360

[8] H.S. Shapiro Notes on a theorem of Baouendi and Rothschild, Exposition. Math., Volume 13 (1995), pp. 247-275

[9] H.S. Shapiro Rate of decay of convolution vs. frequency and sign changes (J.S. Byrnes, ed.), Twentieth Century Harmonic Analysis – a Celebration, Kluwer Academic, Dordrecht, 2001, pp. 383-384

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