Gabor frames with atoms in M^q(R) but not in M^p(R) for any 1≤ p < q ≤ 2

This paper consists of two parts. In the first half, we solve the question raised by Heil as to whether the atom of a Gabor frame must be in Mᵖ(R) for some $1<p<2$. Specifically, for each 0<α β ≤ 1 and 1<q≤ 2 we explicitly construct Gabor frames G(g,α,β) with atoms in Mq(R) but not in Mᵖ(R) for any 1≤ p<q. To construct such Gabor frames, we use box functions as the window functions and show that f = ∑k,n∈ Z f,Mβ nTα k F(χ[0,α]) Mβ nTα k ( F(χ[0,α])) holds for f∈ Mp,q(R) with unconditional convergence of the series for any 0<αβ ≤ 1, 1<p<∞ and 1≤ q<∞. In the second half of this paper, we study two questions related to unconditional convergence of Gabor expansions in modulation spaces. Under the assumption that the window functions are chosen from Mᵖ(R) for some 1≤ p≤ 2, we will prove several equivalent statements that the equation f = ∑k,n∈ Z f, Mβ nTα k γ Mβ nTα k g can be extended from L²(R) to Mq(R) for all f∈ Mq(R) and all p≤ q≤ p' with unconditional convergence of the series. Finally, we characterize all Gabor systems β nTα kg,k∈ Z in Mp,q(R) for any 1≤ p,q<∞ for which f = ∑ f, γk,n Mβ nTα k g with unconditional convergence of the series for all f in Mp,q(R) and all alternative duals \γk,n,n∈ Z of β nTα k g,k∈ Z.