The Minkowski $ ?(x)$ function and Salemʼs problem

Giedrius Alkauskas

doi:10.1016/j.crma.2012.01.012

Mathematical Analysis/Number Theory

The Minkowski

? (x)

function and Salemʼs problem

Presented by: the Editorial Board

Giedrius Alkauskas ¹

¹ Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 Vilnius, Lithuania

Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 137-140.

Abstract
Résumé

R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943) 427–439) asked whether the Fourier–Stieltjes transform of the Minkowski question mark function $? (x)$ vanishes at infinity. In this Note we present several possible approaches towards the solution. For example, we show that this transform satisfies integral and discrete functional equations. Thus, we expect the affirmative answer to Salemʼs problem. At the end of this Note we show that a recent attempt to settle this question (S. Yakubovich, C. R. Acad. Sci. Paris, Ser. I 349 (11–12) (2011) 633–636) is fallacious.

R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943) 427–439) demande si la transformée de Fourier–Stieltjes de la fonction point dʼinterrogation de Minkowski $? (x)$ sʼannule à lʼinfini. Dans cette Note nous présentons plusieurs approches afin de résoudre cette question. Nous montrons par exemple que cette transformée satisfait des équations fonctionnelles discrètes et entières. Ainsi, nous conjecturons une réponse positive au problème de Salem. À la fin de cette Note, nous montrons quʼune tentative récente pour répondre à cette question (S. Yakubovich, C. R. Acad. Sci. Paris, Ser. I 349 (11–12) (2011) 633–636) est en fait incorrecte.

Received: 2011-07-13
Accepted: 2012-01-09
Published online: 2012-01-28

DOI: 10.1016/j.crma.2012.01.012

Author's affiliations:

Giedrius Alkauskas ¹

¹ Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 Vilnius, Lithuania

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     title = {The {Minkowski} $ ?(x)$ function and {Salem's} problem},
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Giedrius Alkauskas. The Minkowski $ ?(x)$ function and Salemʼs problem. Comptes Rendus. Mathématique, Volume 350 (2012) no. 3-4, pp. 137-140. doi : 10.1016/j.crma.2012.01.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.01.012/

References
Cited by

[1] G. Alkauskas The moments of Minkowski question mark function: the dyadic period function, Glasgow Math. J., Volume 52 (2010) no. 1, pp. 41-64

[2] R. Salem On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. Soc., Volume 53 (1943) no. 3, pp. 427-439

[3] N. Temme, Asymptotics of the integral $\int_{0}^{1} \cos (a / x + b x) d x$ , preprint.

[4] G.N. Watson A Treatise on the Theory of Bessel Function, Cambridge University Press, 1995 (reprint of the second (1944) edition)

[5] S. Yakubovich The Fourier–Stieltjes transform of Minkowskiʼs $? (x)$ function and an affirmative answer to Salemʼs problem, C. R. Acad. Sci. Paris, Ser. I, Volume 349 (2011) no. 11–12, pp. 633-636

[6] A. Zygmund Trigonometric Series, vols. I, II, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1988 (reprint of the 1979 edition)

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