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Register a new userBoundary interpolation for slice hyperholomorphic Schur functions
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A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers $kappa_1, ldots, kappa_N$, quaternions $p_1, ldots, p_N$ all of modulus $1$, so that the $2$-spheres determined by each point do not intersect and $p_u eq 1$ for $u = 1,ldots, N$, and quaternions $s_1, ldots, s_N$, we wish to find a slice hyperholomorphic Schur function $s$ so that $$lim_{substack{rrightarrow 1 rin(0,1)}} s(r p_u) = s_uquad {rm for} quad u=1,ldots, N,$$ and $$lim_{substack{rrightarrow 1 rin(0,1)}}frac{1-s(rp_u)overline{s_u}}{1-r}lekappa_u,quad {rm for} quad u=1,ldots, N.$$ Our arguments relies on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.
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In this paper we prove a quaternionic positive real lemma as well as its generalized version, in case the associated kernel has negative squares for slice hyperholomorphic functions. We consider the case of functions with positive real part in the half space of quaternions with positive real part, as well as the case of (generalized) Schur functions in the open unit ball.
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