Boundary interpolation for slice hyperholomorphic Schur functions

Abstract

A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $\hat{I}{º}_1,\dots ,\hat{I}{º}_N$, quaternions $p_1,\dots ,p_N$ all of modulus $1$, so that the $2$-spheres determined by each point do not intersect and $p_u\ne 1$ for $u=1,\dots ,N$, and quaternions $s_1,\dots ,s_N$, we wish to find a slice hyperholomorphic Schur function $s$ so that $$\underset{\begin{array}{c}r\to 1\\ r\in (0,1)\end{array}}{\lim }s(r{p}_u)=s_u\mathrm{for}u=1,\dots ,N,$$ and $$\underset{\begin{array}{c}r\to 1\\ r\in (0,1)\end{array}}{\lim }\frac{1-s(r{p}_u)\overline{s_u}}{1-r}\le \hat{I}{º}_u,\mathrm{for}u=1,\dots ,N.$$ Our arguments relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.