On band circulant matrices in the periodic spline interpolation theory (Q1070467)

The interpolation of a periodic function by a periodic spline of degree n with equal intervals requires the solution of a linear system \(C_ ns=b\), where \(s,b\in {\mathbb{R}}^ N\) and \(C_ n\) is a band circulant matrix of order \(N\geq n\). When n is odd, \(C_ n\) is always invertible, and it is known [\textit{D. Kershaw}, J. Inst. Math. Appl. 20, 227-228 (1977; Zbl 0374.65006)] an explicit bound for \(\| C_ n^{-1}\|\). When n is even, \(C_ n\) is invertible if and only if N is odd. The author extends the method developed by D. Kershaw to the even case. He uses a regular transformation E to obtain \(s=(EC_ n)^{-1}Eb\), decomposes \(EC_ n\) into a product of strictly diagonally dominant circulant tridiagonal matrices and obtains an explicit bound for \(\| (EC_ n)^{-1}\|\) (when n is even and N is \(odd>n)\).