# Interpolation by polynomials with symmetries on the imaginary axis Academic Article

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• Overview
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### 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 …

### publication date

• August 9, 2012