The Littlewood Problem And Problem Of Existence Of Gabor Frames

We consider a basic question in Gabor analysis of classifying all the triples (g,a,b) such that g generates a Weyl-Heisenberg frame with respects to the paramenters a and b. The function g we consider here is x_e, E ∈ R. The cases we study read as follows:((1) We study the case of where {n_i} and . By solving the equivalent Littlewood problem we character for which is a frame.We get full characterization when k ∈ {1,2,3,4} and get some results for k ≥ 5. Then we find some connection between (X_e, a, 1), a < 1 and (X_e, 1,1).(2) By the approach Gu,Hem used to solve the ABC problem, we study the case of {X_E, a, b) where and the case of (g,a,b) ,when g =X_e, where E = [0, c), andc > 0.