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Boundary Interpolation for Slice Hyperholomorphic Schur Functions

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Abstract

A boundary Nevanlinna–Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers \({\kappa_1,\ldots,\kappa_N}\), quaternions p 1,…,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 s 1,…, s N , we wish to find a slice hyperholomorphic Schur function s so that

$$\lim_{\substack{r\rightarrow 1\\ r\in(0,1)}} s(r p_u) = s_u\quad \hbox{for}\ u=1,\ldots, N,$$

and

$$\lim_{\substack{r\rightarrow 1\\ r \in(0,1)}}\frac{1-s(rp_u)\overline{s_u}}{1-r}\le\kappa_u,\quad\hbox{for}\ u=1,\ldots, N.$$

Our arguments rely on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.

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Correspondence to Irene Sabadini.

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D. Alpay thanks the Earl Katz family for endowing the chair which supported his research, and the Binational Science Foundation Grant number 2010117. F. Colombo and I. Sabadini acknowledge the Center for Advanced Studies of the Mathematical Department of the Ben-Gurion University of the Negev for the support and the kind hospitality during the period in which part of this paper has been written. D. P. Kimsey gratefully acknowledges funding from a Kreitman postdoctoral fellowship.

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Abu-Ghanem, K., Alpay, D., Colombo, F. et al. Boundary Interpolation for Slice Hyperholomorphic Schur Functions. Integr. Equ. Oper. Theory 82, 223–248 (2015). https://doi.org/10.1007/s00020-014-2184-3

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