On convergence of oscillatory ergodic Hilbert transforms

Pointwise Convergence of the Ergodic Bilinear Hilbert Transform

Let X = (X,Σ,m, τ) be a dynamical system. We prove that the bilinear series ∑ ′N n=−N f(τnx)g(τ−nx) n converges almost everywhere for each f, g ∈ L(X). We also give a proof along the same lines of Bourgain’s analog result for averages.

On the convergence of the rotated one-sided ergodic Hilbert transform

Sufficient conditions have been given for the convergence in norm and a.e. of the ergodic Hilbert transform ([11], [5], [6]). Here we apply these conditions to the rotated ergodic Hilbert transform ∑ ∞ n=1 λ n n T f , where λ is a complex number of modulus 1. When T is a contraction in a Hilbert space, we show that the logarithmic Hausdorff dimension of the set of λ’s for which this series does...

Convergence Accelerator Approach to the Numerical Evaluation of Hilbert Transforms Based on Expansions in Hermite Functions

The evaluation of the Hilbert transform of a function is a very important problem that appears widely in science and engineering, specially in signal analysis. In this work, the numerical evaluation of a class of Hilbert transforms using an expansion in terms of Hermite functions, eigenfunctions of the Fourier transform, is performed. Judicious selection of convergence accelerators allows for t...

Two-dimensional Hilbert Transforms'

Change of variable formulæ for regularizing slowly decaying and oscillatory Cauchy and Hilbert transforms

Formulæ are derived for expressing Cauchy and Hilbert transforms of a function f in terms of Cauchy and Hilbert transforms of f(xr). When r is an integer, this corresponds to evaluating the Cauchy transform of f(xr) at all choices of z1/r. Related formuæ for rational r result in a reduction to a generalized Cauchy transform living on a Riemann surface, which in turn is reducible to the standard...