Interpolation By Polynomials With Symmetries

Authors

Daniel Alpay, Chapman UniversityFollow
Izchak Lewkowicz, Ben Gurion University of the Negev

Document Type

Article

Publication Date

2014

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 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.

Comments

NOTICE: this is the author’s version of a work that was accepted for publication in Linear Algebra and its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Linear Algebra and its Applications, volume 456, in 2014. DOI: 10.1016/j.laa.2012.10.037

The Creative Commons license below applies only to this version of the article.

Recommended Citation

D. Alpay and I. Lewkowicz. Interpolation by polynomials with symmetries. Linear Algebra and its Applications, vol. 456 (2014), 64-81.

Peer Reviewed

1

Copyright

Elsevier

Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 License.