ON HILBERT TRANSFORM OF GABOR AND WILSON SYSTEMS

Abstract

Given that the Gabor system {EmbTnag}m,n∈ℤ is a Gabor frame for L2(ℝ), a sufficient condition is obtained for the Gabor system {EmbTnaHg}m,n∈ℤ to be a Gabor frame, where Hg denotes the Hilbert transform of g ∈ L2(ℝ). It is proved that the Hilbert transform operator and the frame operator for the Gabor Bessel sequence {EmbTnag}m,n∈ℤ commute with each other under certain conditions. Also, a sufficient condition is obtained for the Wilson system [Formula: see text] to be a Wilson frame given that [Formula: see text] is a Wilson frame. Finally, we obtain conditions under which the Hilbert transform operator and the frame operator for the Wilson Bessel sequence [Formula: see text] commute with each other.