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]r,s=0k−1 , where k > 1, C0, C1, …, Ck−1∈Cd1×d2 , α ∈ {1,2, … ,k −1}, gcd(α,k) = 1, and ∥ ⋅ ∥ is the Frobenius norm.
Minimization problems for (R,S)-symmetric and (R,S)-skew symmetric matrices
Let R∈Cm×m and S∈Cn×n be nontrivial involutions; thus R=R−1≠±I and S=S−1≠±I. We say that A∈Cm×n is (R,S)-symmetric ((R,S)-skew symmetric) if RAS=A (RAS=−A). Let S be the class of m×n (R,S)-symmetric matrices or the class of m×n (R,S)-skew symmetric matrices. Let Z∈Cn×q and W∈Cm×q. We study the following problems: 
(i) 
Give necessary and sufficient conditions for the existence of an A∈S such that AZ=W, and find all such matrices if the conditions are met. 
 
(ii) 
Find σ(Z,W)=minA∈S∥AZ−W∥ and characterize the class S(Z,W)={A∈S|∥AZ−W∥=σ(Z,W)}. 
 
(iii) 
If B∈Cm×n is arbitrary, find σ(Z,W,B)=minA∈S(Z,W)∥A−B∥ and find A∈S(Z,W) such that ∥A−B∥=σ(Z,W,B). 
 
 
 
 
We obtain explicit formulas for σ(Z,W), σ(B,Z,W), and all the matrices in question.
A new class of numerical sequences and its applications to sine and cosine series
First, it is proved that the class of classical quasi-monotone sequences is not comparable to the newly defined class of sequences of ‘rest bounded variation’. Considering this result, we prove three sample theorems for sequences of rest bounded variation, being analogues of the theorems proved earlier for monotone or quasi-monotone sequences. One of them gives a partial answer to a question raised by R. P. Boas Jr.