An Interplay between Gabor and Wilson Frames

Abstract

Wilson frames{ψjk:w0,w-1∈L2(ℝ)}j∈ℤ,k∈ℕ0as a generalization of Wilson bases have been defined and studied. We give necessary condition for a Wilson system to be a Wilson frame. Also, sufficient conditions for a Wilson system to be a Wilson Bessel sequence are obtained. Under the assumption that the window functionsw0andw-1for odd and even indices ofjare the same, we obtain sufficient conditions for a Wilson system to be a Wilson frame (Wilson Bessel sequence). Finally, under the same conditions, a characterization of Wilson frame in terms of Zak transform is given.

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].…”

Wilson basis expansions of electromagnetic wavefields: a suitable framework for fiber optics

“…For instance, studies that focus on its connection with Gabor frames (Bownik et al 2017;Kaushik and Panwar 2013), construction of Wilson bases with arbitrary shapes (Bittner 1999;Chui and Shi 2000), or studies of discretized Wilson bases (Bolcskei et al 1996;Kutyniok and Strohmer 2005). So far, the use of Wilson bases in electromagnetics has only been advocated by Arnold (2002).…”