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Inverse problems for unilevel block α-circulants


  • Dedicated to Professor Biswa Nath Datta.

Correspondence to: W. Trench, 659 Hopkinton Road, Hopkinton, NH 03229 USA.



We consider the following inverse problems for the class math formula of unilevel block α-circulants math formula, where k > 1, C0, C1, …, math formula, α ∈ {1,2, … ,k −1}, gcd(α,k) = 1, and ∥ ⋅ ∥ is the Frobenius norm.

Problem 1 Find necessary and sufficient conditions on math formula and math formula for the existence of math formula such that CZ = W, and find all such C if the conditions are satisfied.

Problem 2 For arbitrary math formula and math formula, find

display math

characterize the class

display math

and find C in this class with minimum norm.

Problem 3 If math formula is given, find

display math

and find math formula such that ∥ CA ∥ = σα(Z,W,A).

We also consider slightly modified problems for the case where gcd(α,k) > 1. Copyright © 2011 John Wiley & Sons, Ltd.