Inverse problems for unilevel block α-circulants

We consider the following inverse problems for the class Cα of unilevel block α-circulants C = [Cs−αr ] k−1 r,s=0, where k > 1, C0, C1, . . . , Ck−1 ∈ Cd1×d2 , α ∈ {1, 2, . . . , k − 1}, gcd(α, k) = 1, and ‖ · ‖ is the Frobenius norm. Problem 1 Find necessary and sufficient conditions on Z ∈ Ckd2×h and W ∈ C1 for the existence of C ∈ Cα such that CZ = W , and find all such C if the conditions are satisfied. Problem 2 For arbitrary Z ∈ C2 and W ∈ C1, find σα(Z,W ) = min C∈Cα ‖CZ −W‖, characterize the class Mα(Z,W ) = { C ∈ Cα ∣∣ ‖CZ −W‖ = σα(Z,W ) } , and find C in this class with minimum norm. Problem 3 If A ∈ Cα is given, find σα(Z,W,A) = min C∈Mα(Z,W ) ‖C −A‖ and find C ∈ Mα(Z,W ) such that ‖C −A‖ = σα(Z,W,A). We also consider slightly modified problems for the case where gcd(α, k) > 1. MSC: 15A09; 15A15; 15A18; 15A99