Cantor uniqueness and multiplicity along subsequences

G. Kozma ; A. M. Olevskiǐ

Algebra i analiz, Tome 32 (2020) no. 2, pp. 85-106.

Résumé

We construct a sequence $ c_{l}\to 0$ such that the trigonometric series $ \sum c_{l}e^{ilx}$ converges to zero everywhere on a subsequence $ n_{k}$. We show, for any such series, that the $ n_{k}$ must be very sparse, and that the support of the related distribution must be quite large.

Détail

Keywords: trigonometric series, localization principle, uniqueness.

Bibliographie

[1] Arutyunyan F. G., Talalyan A. A., “O edinstvennosti ryadov po sistemam Khaara i Uolsha”, Izv. AN SSSR. Ser. mat., 28:6 (1964), 1391–1408 | Zbl

[2] Ash Marshall J., “A new proof of uniqueness for multiple trigonometric series”, Proc. Amer. Math. Soc., 107:2 (1989), 409–410 | DOI | MR | Zbl

[3] Bari N. K., “O vsyudu skhodyaschikhsya k nulyu podposledovatelnostyakh chastnykh summ trigonometricheskogo ryada”, Izv. AN SSSR. Ser. mat., 24:4 (1960), 531–548 | Zbl

[4] Bary N. K., A treatise on trigonometric series, v. I, II, A Pergamon Press Book The Macmillan Co., New York, 1964 | MR | Zbl

[5] Benedicks M., “The support of functions and distributions with a spectral gap”, Math. Scand., 55:2 (1984), 285–309 | DOI | MR | Zbl

[6] Cantor G., “Beweis, dass eine für jeden reellen Werth von $x$ durch eine trigonometrische Reihe gegebene Function $f(x)$ sich nur auf eine einzige Weise in dieser Form darstellen lässt”, J. Reine Angew. Math., 72 (1870), 139–142 | MR

[7] Cooke R., “Uniqueness of trigonometric series and descriptive set theory, 1870–1985”, Arch. Hist. Exact Sci., 45:4 (1993), 281–334 | DOI | MR | Zbl

[8] Falconer K., Fractal geometry, Mathematical foundations and applications, Third ed., John Wiley Sons, Ltd., Chichester, 2014 | MR | Zbl

[9] Kahane J.-P., Salem R., Ensembles parfaits et séries trigonométriques, 2nd ed., Hermann, Paris, 1994 | MR

[10] Katznelson Y., An introduction to harmonic analysis, Cambridge Math. Library, Third ed., Cambridge Univ. Press, Cambridge, 2004 | MR | Zbl

[11] Kechris A. S., Louveau A., Descriptive set theory and the structure of sets of uniqueness, London Math. Soc. Lecture Note Ser., 128, Cambridge Univ. Press, Cambridge, 1987 | MR | Zbl

[12] Kozlov V. Ya., “O polnykh sistemakh ortogonalnykh funktsii”, Mat. sb., 26:3 (1950), 351–364 | Zbl

[13] Olevskii A. M., “Modifikatsiya funktsii i ryady Fure”, Uspekhi mat. nauk, 40:3(243) (1985), 157–193 | MR

[14] Ransford T., Potential theory in the complex plane, London Math. Soc. Student Texts, 28, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl

[15] Skvortsov V. A., “Primer ryada Uolsha so vsyudu skhodyascheisya k nulyu podposledovatelnostyu chastichnykh summ”, Mat. sb., 97:4 (1975), 517–539 | MR | Zbl

[16] Talalyan A. A., “Predstavlenie izmerimykh funktsii ryadami”, Uspekhi mat. nauk, 15:5 (1960), 77–141 | MR | Zbl

[17] Ulyanov P. L., “Reshennye i nereshennye problemy teorii trigonometricheskikh i ortogonalnykh ryadov”, Uspekhi mat. nauk, 19:1 (1964), 3–69 | MR

[18] Zygmund A., Trigonometric series, With a foreword by Robert A. Fefferman, v. I, II, Cambridge Mathematical Library, Third edition, Cambridge Univ. Press, Cambridge, 2002 | MR | Zbl