The Fourier-Stieltjes transform of Minkowski's ?(x) function and an affirmative answer to Salem's problem

TL;DR

This paper provides a definitive affirmative answer to Salem's long-standing question about the vanishing of the Fourier-Stieltjes transform of Minkowski's question mark function at infinity, using properties of the Kontorovich-Lebedev transform.

Contribution

It offers a correct, alternative proof that the Fourier-Stieltjes transform of Minkowski's question mark function vanishes at infinity, resolving Salem's problem and its generalizations.

Findings

The Fourier-Stieltjes transform of Minkowski's question mark function vanishes at infinity.

The paper presents a new proof using the Kontorovich-Lebedev transform.

Salem's problem is conclusively solved with this approach.

Abstract

By using structural and asymptotic properties of the Kontorovich-Lebedev transform associated with Minkowski's question mark function, we give an affirmative answer to the question posed by R. Salem (Trans. Amer. Math. Soc., 53 (3), (1943), p. 439) whether its Fourier-Stieltjes transform vanishes at infinity. This paper is substituted with a correct alternative and affirmative solution of Salem's problem and its generalization for derivatives of any order of the Fourier-Stieltjes transform of the Minkowski'i question mark function. The question is finally solved.