Characteristic Functions As Gabor Window Functions

The theory of frames for a Hilbert space plays a fundamental role in signal processing, image processing, data compression, robust data transmission and more. In the theory of frames, Gabor analysis is one of the extensive realms which need most to develop with thorough research. A basic question in Gabor analysis is to determine whether {E_mbT_nag}_m,(n∈Z) is a frame, with respect to given g∈L²(R) and parameters a,b e R. We denote {E_mbT_nag}_m,(n∈Z) as (g,a,b) for short. In this paper, we focus on E which is a non-null bounded measurable subset in R, and determine the sufficient and necessary conditions under which (X_E, a, b) is a frame . We defined a functin (?), and say when (?)degree of overlap in E is N, then we focus on two problems as follow:1. We study the sufficient and necessary conditions under which (X_E, a, b) is a frame, when (?).2. We study the sufficient and necessary conditions under which (X_E, a, b) is a frame, when (?).