Interpolation Khrushchev-type formulas for structured operators, inequalities and asymptotic relations

We show that interpolation results in the $S$-nodes theory may be considered as Khrushchev-type formulas. If separation of the well-known Verblunsky (Schur) coefficients occurs in Khrushchev formulas, the separation of the so the called new Verblunsky-type coefficients occurs in the interpolation formulas of the $S$-nodes theory. General asymptotic inequalities (and equalities) for the $S$-nodes, with application to the block Hankel matrices, are derived using this approach. Another asymptotic inequality, needed in the proofs and important in itself, is derived in Appendix B.