The Fourier-Stieltjes transform of Minkowski's ?(x) function and an affirmative answer to Salem's problem
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.
Keywords
Cite
@article{arxiv.1103.3979,
title = {The Fourier-Stieltjes transform of Minkowski's ?(x) function and an affirmative answer to Salem's problem},
author = {Semyon Yakubovich},
journal= {arXiv preprint arXiv:1103.3979},
year = {2011}
}
Comments
This paper has been withdrawn by the author, because the main result is based on Naylor's asymptotic formula for the Kontorovich-Lebedev transform at infinity, whose proof has a gap for extreme values of a parameter under required continuity condition and needs perhaps more strong condition of analyticity in some sector of complex plane