On Hilbert Transform of Gabor and Wilson Systems

Abstract

Given that the Gabor system {E mb Tnag} m,n∈Z is a Gabor frame for L 2 (R), a sufficient condition is obtained for the Gabor system {E mb TnaHg} m,n∈Z to be a Gabor frame, where Hg denotes the Hilbert transform of g ∈ L 2 (R). It is proved that the Hilbert transform operator and the frame operator for the Gabor Bessel sequence {E mb Tnag} m,n∈Z commute with each other under certain conditions. Also, a sufficient condition is obtained for the Wilson system {ψ k,n Hg} k∈Z n∈N 0 to be a Wilson frame given that {ψ k,n g} k∈Z n∈N 0 is a Wilson frame. Finally, we obtain conditions under which the Hilbert transform operator and the frame operator for the Wilson Bessel sequence {ψ k,n g} k∈Z n∈N 0 commute with each other.Example 3.1. Consider the function g = cos( 2πx 3 )χ [0, 3 4 ] ∈ L 2 (R), then it is easy to observe that {E mb T na g} m,n∈Z and {E mb T na Hg} m,n∈Z are both Gabor frames for 0 < a < 3 4 and 0 < b ≤ 4 3 . It is natural to raise the following general question.Problem: Let g ∈ L 2 (R) be such that {E mb T na g} m,n∈Z is a Gabor frame. Is {E mb T na Hg} m,n∈Z a Gabor frame?The answer to this problem is negative.Example 3.2. If g = χ [0,1] , then {E m T n g} m,n∈Z is a Gabor frame but {E m T n Hg} m,n∈Z is not a Gabor frame.

On Discrete Time Wilson Systems

“…Wilson [3,4] suggested a system of functions which are localized around the positive and negative frequency of the same order. Based on the Wilson systems, Wilson frames for L 2 (R) were introduced and studied in [17][18][19][20]. In this article, discrete time Wilson frames (DTWF) are defined and their relationship with discrete time Gabor frames is investigated.…”

“…Motivated by the fact that one has different trigonometric functions for odd and even indices, Bittner [11,16] considered Wilson bases introduced by Daubechies et.al [5] with nonsymmetrical window functions for odd and even indices. is generalized system of Bittner was later studied extensively by Kaushik and Panwar [17][18][19] and Jarrah and Panwar [20].…”