Boundary interpolation for slice hyperholomorphic Schur functions

2014/04/13 by Abu-Ghanem, K., Alpay, D., Colombo, F. +2
#Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1404.3352

Abstract

A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers κ₁, …, κ_N, quaternions p₁, …, p_N all of modulus 1, so that the 2-spheres determined by each point do not intersect and p_u ≠ 1 for u = 1,…, N, and quaternions s₁, …, s_N, we wish to find a slice hyperholomorphic Schur function s so that lim_\substackr→ 1
r∈(0,1) s(r p_u) = s_u \rm for u=1,…, N, and lim_\substackr→ 1
r∈(0,1)\frac1-s(rp_u)s_u1-r≤κ_u, \rm for u=1,…, N. Our arguments relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.

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