On the affirmative solution to Salem's problem

TL;DR

This paper confirms that certain Fourier-Stieltjes coefficients related to the Minkowski question mark function tend to zero at infinity, using classical analysis and special functions, and explores related asymptotic behaviors.

Contribution

It provides an affirmative solution to a Salem-type problem concerning the asymptotic behavior of Fourier-Stieltjes transforms of the Minkowski question mark function.

Findings

Fourier-Stieltjes coefficients vanish at infinity.

Derived asymptotic relations for coefficients of powers of the function.

Solved a longstanding problem in harmonic analysis related to the Minkowski question mark function.

Abstract

The Salem problem to verify whether Fourier-Stieltjes coefficients of the Minkowski question mark function vanish at infinity is solved recently affirmatively. In this paper by using methods of classical analysis and special functions we solve a Salem-type problem about the behavior at infinity of a linear combination of the Fourier-Stieltjes transforms. Moreover, as a consequence of the Salem problem, some asymptotic relations at infinity for the Fourier-Stieltjes coefficients of a power $m\in \mathbb{N}$ of the Minkowski question mark function are derived.