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

TL;DR

This paper explores interpolation formulas in the $S$-nodes theory as Khrushchev-type formulas, deriving new asymptotic inequalities for structured operators and applying these results to block Hankel matrices.

Contribution

It introduces a novel perspective linking $S$-nodes interpolation formulas with Khrushchev-type formulas, leading to new asymptotic inequalities for structured operators.

Findings

Derived new asymptotic inequalities for $S$-nodes.

Established connections between $S$-nodes interpolation and Khrushchev formulas.

Applied results to block Hankel matrices.

Abstract

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.