Everywhere divergence of the one-sided ergodic Hilbert transform

Classification of operator algebras: complexity, rigidity, and dynamics

13 April 14:00 - 15:00

Jörg Schmeling - Lund University

For a given number alpha in (0; 1) and a 1-periodic function f, we study the convergence of the series \sum n=1^\infty f(x+n )/n, called one-sided Hilbert transform relative to the rotation x -> x+alpha mod 1. Among others, we prove that for any non-polynomial function of class C^2 having Taylor- Fourier series (i.e. Fourier coefficients vanish on Z_-), there exists an irrational number alpha (actually a residual set of alphas) such that the series diverges for all x. We also prove that for any irrational number alpha , there exists a continuous function f such that the series diverges for all x. The convergence of general series \sum n=1^\infty a_nf(x+n) is also discussed in different cases involving the diophantine property of the number alpha and the regularity of the function f.
Marius Dadarlat
Purdue University
Søren Eilers
University of Copenhagen
Asger Törnquist
University of Copenhagen


Søren Eilers


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