A sharp one-sided bound for the Hilbert transform

Everywhere divergence of the one-sided ergodic Hilbert transform for circle rotations by Liouville numbers

We prove some results on the behavior of infinite sums of the form ∑ f ◦ T(x) 1 n , where T : S → S is an irrational circle rotation and f is a mean-zero function on S. In particular, we show that for a certain class of functions f , there are Liouville α for which this sum diverges everywhere and Liouville α for which the sum converges everywhere.

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...

One-Sided Stability of Medial Axis Transform

A Sharp Estimate for the Hilbert Transform along Finite Order Lacunary Sets of Directions

Let D be a nonnegative integer and Θ ⊂ S1 be a lacunary set of directions of order D. We show that the Lp norms, 1 < p < ∞, of the maximal directional Hilbert transform in the plane HΘ f (x ) B sup v ∈Θ p.v. ∫ R f (x + tv ) dt t , x ∈ R 2, are comparable to (log #Θ) 2 . For vector elds vD with range in a lacunary set of of order D and generated using suitable combinations of truncations of Lips...

One-Dimensional Processing for Edge Detection using Hilbert Transform

This paper presents a new method for edge detection using one-dimensional processing. The Discrete Hilbert Transform of a Gaussian function is used as an edge detection lter. The image is smoothed using 1-D Gaussian along the horizontal (or vertical) scan lines to reduce noise. Detection lter is then used in the orthogonal direction, i.e., along vertical (or horizontal) scan lines to detect the...