[Sur les changements du signe de distributions tempérées ayant une lacune spectrale à l'origine]
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.
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@article{CRMATH_2003__336_4_325_0,
author = {Ostrovskii, Iossif and Ulanovskii, Alexander},
title = {On sign changes of tempered distributions having a spectral gap at the origin},
journal = {Comptes Rendus. Math\'ematique},
pages = {325--330},
publisher = {Elsevier},
volume = {336},
number = {4},
year = {2003},
doi = {10.1016/S1631-073X(03)00036-0},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00036-0/}
}
TY - JOUR AU - Ostrovskii, Iossif AU - Ulanovskii, Alexander TI - On sign changes of tempered distributions having a spectral gap at the origin JO - Comptes Rendus. Mathématique PY - 2003 SP - 325 EP - 330 VL - 336 IS - 4 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(03)00036-0/ DO - 10.1016/S1631-073X(03)00036-0 LA - en ID - CRMATH_2003__336_4_325_0 ER -
%0 Journal Article %A Ostrovskii, Iossif %A Ulanovskii, Alexander %T On sign changes of tempered distributions having a spectral gap at the origin %J Comptes Rendus. Mathématique %D 2003 %P 325-330 %V 336 %N 4 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(03)00036-0/ %R 10.1016/S1631-073X(03)00036-0 %G en %F CRMATH_2003__336_4_325_0
Ostrovskii, Iossif; Ulanovskii, Alexander. On sign changes of tempered distributions having a spectral gap at the origin. Comptes Rendus. Mathématique, Tome 336 (2003) no. 4, pp. 325-330. doi : 10.1016/S1631-073X(03)00036-0. http://www.numdam.org/articles/10.1016/S1631-073X(03)00036-0/
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