Inverses of cyclic band matrices and the convergence of interpolation processes for derivatives of periodic interpolation splines

Entries of matrices inverse to cyclic band ones, as well as norms of the inverse matrices, are estimated. The estimates obtained are used to specify conditions under which interpolation processes with periodic odd degree splines converge for different derivatives. In particular, we give a positive solution to the C. de Boor problem on unconditional convergence of one of two middle derivatives without imposing any restrictions on a grid in the periodic case.