On some Rajchman measures and a solution to Salem's problem revisited

TL;DR

This paper constructs special Rajchman measures using Fourier transform properties, proves that the Minkowski question mark function's Fourier-Stieltjes coefficients vanish at infinity, and extends this to derivatives of all orders.

Contribution

It provides a rigorous proof that the Fourier-Stieltjes coefficients of Minkowski's question mark function tend to zero, confirming Salem's problem and generalizing to derivatives.

Findings

Fourier-Stieltjes coefficients of ?(x) vanish at infinity.

Minkowski's question mark function belongs to a class of Rajchman measures.

Derivatives of Fourier-Stieltjes transforms also tend to zero.

Abstract

We construct certain Rajchman measures by using integrability properties of the Fourier and Fourier-Stieltjes transforms. Further we show that Minkowski's question mark function ?(x), which is a singular monotone function, belongs to one of these classes. This circumstance leads us to a rigorous proof of the problem posed by R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943), p. 439) whether Fourier-Stieltjes coefficients of this function vanish at infinity and whose affirmative solution was recently announced by the author in [13]. Moreover, we generalize this problem, proving that derivatives of any order of the Fourier-Stieltjes transform with respect to the measure ?(x) tend to zero as well.