虚轴上对称多项式插值
Interpolation by polynomials with symmetries on the imaginary axis
摘要：
在这里，我们专门的标准矩阵值多项式插值的情况下，在虚轴上的插值多项式承认各种对称性：半正定，斜埃尔米特，$J$ -埃尔米特，汉密尔顿和其他。该过程是三个阶段，说明了通过的情况下，在$i\R$的插值多项式是半正定的。首先，以加倍次数为代价，得到一个极小次插值多项式$P(s)$，它在$i\R$上是Hermite的.然后，我们找到所有的多项式$Ψ(s)$，消失的插值点是半正定的$i\R$。最后，利用正半定矩阵的集合是埃尔米特矩阵的凸子锥的事实，可以计算最小标量$\hatβ\geq 0$，使得$P(s)+βΨ(s)$满足所有$β\geq\hatβ$的所有插值约束。这种方法，然后适应的情况下，当家庭的插值多项式不是凸的。只要是凸的，我们就参数化所有的最小次插值多项式。
英文摘要：
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 $\hatβ\geq 0$ so that $P(s)+βΨ(s)$ satisfies all interpolation constraints for all $β\geq\hatβ$. 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.
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