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 Ψ(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 βˆ ≥ 0 so that P(s)+ βΨ(s) satisfies all interpolation constraints for all β ≥ βˆ .
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.
D. Alpay and I. Lewkowicz. Interpolation by polynomials with symmetries. Linear Algebra and its Applications, vol. 456 (2014), 64-81.
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