Geodesic
Absolutely Convergent Fourier Series. An Improvement of the Beurling--Helson Theorem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 2, pp. 52-65.

Voir la notice de l'article provenant de la source Math-Net.Ru

We consider the space $A(\mathbb T)$ of all continuous functions $f$ on the circle $\mathbb T$ such that the sequence of Fourier coefficients $\widehat{f}=\{\widehat{f}(k),\,k\in\mathbb Z\}$ belongs to $l^1(\mathbb Z)$. The norm on $A(\mathbb T)$ is defined by $\|f\|_{A(\mathbb T)}=\|\widehat{f}\|_{l^1(\mathbb Z)}$. According to the well-known Beurling–Helson theorem, if $\varphi\colon \mathbb T\to\mathbb T$ is a continuous mapping such that $\|e^{in\varphi}\|_{A(\mathbb T)}=O(1)$, $n\in\mathbb Z$, then $\varphi$ is linear. It was conjectured by Kahane that the same conclusion about $\varphi$ is true under the assumption that $\|e^{in\varphi}\|_{A(\mathbb T)}=o(\log |n|)$. We show that if $\|e^{in\varphi}\|_{A(\mathbb T)}=o((\log\log |n|/\log\log\log |n|)^{1/12})$, then $\varphi$ is linear.
Keywords: absolutely convergent Fourier series, Beurling–Helson theorem. 
@article{FAA_2012_46_2_a5,
     author = {V. V. Lebedev},
     title = {Absolutely {Convergent} {Fourier} {Series.} {An} {Improvement} of the {Beurling--Helson} {Theorem}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {52--65},
     publisher = {mathdoc},
     volume = {46},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2012_46_2_a5/}
}
TY  - JOUR
AU  - V. V. Lebedev
TI  - Absolutely Convergent Fourier Series. An Improvement of the Beurling--Helson Theorem
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2012
SP  - 52
EP  - 65
VL  - 46
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2012_46_2_a5/
LA  - ru
ID  - FAA_2012_46_2_a5
ER  - 
%0 Journal Article
%A V. V. Lebedev
%T Absolutely Convergent Fourier Series. An Improvement of the Beurling--Helson Theorem
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2012
%P 52-65
%V 46
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2012_46_2_a5/
%G ru
%F FAA_2012_46_2_a5
V. V. Lebedev. Absolutely Convergent Fourier Series. An Improvement of the Beurling--Helson Theorem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 2, pp. 52-65. http://geodesic.mathdoc.fr/item/FAA_2012_46_2_a5/

[1] A. Beurling, H. Helson, “Fourier-Stieltjes transforms with bounded powers”, Math. Scand., 1 (1953), 120–126 | DOI | MR | Zbl

[2] Zh.-P. Kakhan, Absolyutno skhodyaschiesya ryady Fure, Mir, M., 1976

[3] J.-P. Kahane, “Quatre leçons sur les homéomorphismes du cercle et les séries de Fourier”, Topics in Modern Harmonic Analysis, Vol. II, Ist. Naz. Alta Mat. Francesco Severi, Roma, 1983, 955–990 | MR

[4] Z. L. Leibenzon, “O koltse funktsii s absolyutno skhodyaschimisya ryadami Fure”, UMN, 9:3(61) (1954), 157–162 | MR

[5] J.-P. Kahane, “Sur certaines classes de séries de Fourier absolument convergentes”, J. Math. Pures Appl., 35:3 (1956), 249–259 | MR | Zbl

[6] V. V. Lebedev, “Diffeomorfizmy okruzhnosti i teorema Berlinga–Khelsona”, Funkts. analiz i ego pril., 36:1 (2002), 30–35 | DOI | MR | Zbl

[7] V. V. Lebedev, “Kolichestvennye otsenki v teoremakh tipa teoremy Berlinga–Khelsona”, Matem. sb., 201:12 (2010), 103–130 | DOI | MR

[8] V. V. Lebedev, “Otsenki v teoremakh tipa teoremy Berlinga–Khelsona. Mnogomernyi sluchai”, Matem. zametki, 90:3 (2011), 394–407 | DOI | MR | Zbl

[9] J.-P. Kahane, “Transformées de Fourier des fonctions sommables”, Proc. Internat. Congr. Math. (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, 114–131 | MR

[10] B. Green, S. Konyagin, “On the Littlewood problem modulo a prime”, Canad. J. Math., 61:1 (2009), 141–164 | DOI | MR | Zbl

[11] E. M. Stein, R. Shakarchi, Fourier Analysis: An Introduction, Princeton Lectures in Analysis, I, Princeton Univ. Press, Princeton, NJ, 2003 | MR

[12] V. Shmidt, Diofantovy priblizheniya, Mir, M., 1983 | MR

[13] R. Larsen, An Introduction to the Theory of Multipliers, Springer-Verlag, Berlin–Heidelberg–New York, 1971 | MR | Zbl

[14] M. M. Postnikov, Differentsialnaya geometriya, Lektsii po geometrii. Semestr IV, Nauka, M., 1988 | MR | Zbl

[15] T. Sanders, “The Littlewood–Gowers problem”, J. Anal. Math., 101:1 (2007), 123–162 | DOI | MR | Zbl