Interpolation by polynomials with symmetries on the imaginary axis

TL;DR

This paper develops a method for matrix-valued polynomial interpolation with symmetry constraints on the imaginary axis, including positive semidefinite and Hermitian properties, with applications to minimal degree solutions.

Contribution

It introduces a three-stage procedure for constructing symmetric interpolating polynomials, including convex and non-convex cases, with a focus on positive semidefinite constraints on the imaginary axis.

Findings

Method for constructing minimal degree symmetric interpolants

Parameterization of all minimal degree solutions in convex cases

Adaptation to non-convex interpolation families

Abstract

We here specialize the standard matrix-valued polynomial interpolation to the case where on the imaginary axis the interpolating polynomials admit various symmetries: Positive semidefinite, Skew-Hermitian, $J$-Hermitian, Hamiltonian and others. The procedure is comprized of three stages, illustrated through the case where on $i\R$ the interpolating polynomials are to be positive semidefinite. We first, on the expense of doubling the degree, obtain a minimal degree interpolating polynomial $P(s)$ which on $i\R$ is Hermitian. Then we find all polynomials $\Psi(s)$, vanishing at the interpolation points which are positive semidefinite on $i\R$. Finally, using the fact that the set of positive semidefinite matrices is a convex subcone of Hermitian matrices, one can compute the minimal scalar $\hat{\beta}\geq 0$ so that $P(s)+\beta\Psi(s)$ satisfies all interpolation constraints for all $\beta\geq\hat{\beta}$. This approach is then adapted to cases when the family of interpolating polynomials is not convex. Whenever convex, we parameterize all minimal degree interpolating polynomials.