Polynomial interpolation over quaternions

Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated "on the left" and "on the right". If the interpolation problem involves interpolation conditions of the same (left or right) type, the results are very much similar to the complex case: a consistent problem has a unique solution of a low degree (less than the number of interpolation conditions imposed), and the solution set of the homogeneous problem is an ideal in the ring ${\mathbb H}[z]$. The problem containing both "left" and "right" interpolation conditions is quite different: there may exist infinitely many low-degree solutions and the solution set of the homogeneous problem is a quasi-ideal in ${\mathbb H}[z]$

An algorithm for finding low degree rational solutions to the Schur coefficient problem

We present an algorithm producing all rational functions $f$ with prescribed $n+1$ Taylor coefficients at the origin and such that $\|f\|_\infty\le 1$ and $\deg f\le k$ for every fixed $k\ge n$. The case where $k<n$ is also discussed

Confluent Vandermonde matrices, divided differences, and Lagrange-Hermite interpolation over quaternions

We introduce the notion of a confluent Vandermonde matrix with quaternion entries and discuss its connection with Lagrange-Hermite interpolation over quaternions. Further results include the formula for the rank of a confluent Vandermonde matrix, the representation formula for divided differences of quaternion polynomials and their extensions to the formal power series setting

Pick matricies and quaternionic power series

It is well known that a non-constant complex-valued function $f$ defined on the open unit disk $\mathbb D$ is an analytic self-mapping of \D if and only if Pick matrices $\left[ (1-f(z_i)\overline{f(z_j)})/(1-z_i\overline{z}_j)\right]_{i,j=1}^n$ are positive semidefinite for all choices of finitely many points z_i\in\D. A stronger version of the "if" part was established by Alan Hindmarsh: if all $3\times 3$ Pick matrices are positive semidefinite, then $f$ is an analytic self-mapping of $\mathbb D$. In this paper, we extend this result to the non-commutative setting of power series over quaternions

Zeros and factorizations of quaternion polynomials: the algorithmic approach

It is known that polynomials over quaternions may have spherical zeros and isolated left and right zeros. These zeros along with appropriately defined multiplicities form the zero structure of a polynomial. In this paper, we equivalently describe the zero structure of a polynomial in terms of its left and right spherical divisors as well as in terms of left and right indecomposable divisors. Several algorithms are proposed to find left/right zeros and left/right spherical divisors of a quaternion polynomial, to construct a polynomial with prescribed zero structure and more generally, to construct the least left/right common multiple of given polynomials. Similar questions are briefly discussed in the setting of quaternion formal power series

A uniqueness result on boundary interpolation

Let $f$ be an analytic function mapping the unit disk \D to itself. We give necessary and sufficient conditions on the local behavior of $f$ near a finite set of boundary points that requires $f$ to be a finite Blaschke product

The boundary analog of the Carath\'eodory-Schur interpolation problem

Characterization of Schur-class functions (analytic and bounded by one in modulus on the open unit disk) in terms of their Taylor coefficients at the origin is due to I. Schur. We present a boundary analog of this result: necessary and sufficient conditions are given for the existence of a Schur-class function with the prescribed nontangential boundary expansion $f(z)=s_0+s_1(z-t_0)+\ldots+s_N(z-t_0)^N+o(|z-t_0|^N)$ at a given point $t_0$ on the unit circle

On the Sylvester matrix equation over quaternions

The Sylvester equation $AX-XB=C$ is considered in the setting of quaternion matrices. Conditions that are necessary and sufficient for the existence of a unique solution are well-known. We study the complementary case where the equation either has infinitely many solutions or does not have solutions at all. Special attention is given to the case where $A$ and $B$ are respectively, lower and upper triangular two-diagonal matrices (in particular, if $A$ and $B$ are Jordan blocks

Boundary Nevanlinna--Pick interpolation problems for generalized Schur functions

Three boundary Nevanlinna-Pick interpolation problems at finitely many points are formulated for generalized Schur functions. For each problem, the set of all solutions is parametrized in terms of a linear fractional transformation with a Schur class parameter

The higher order Carath\'eodory--Julia theorem and related boundary interpolation problems

The higher order analogue of the classical Carath\'eodory-Julia theorem on boundary angular derivatives has been obtained in \cite{bk3}. Here we study boundary interpolation problems for Schur class functions (analytic and bounded by one in the open unit disk) motivated by that result