Boundary Interpolation for Slice Hyperholomorphic Schur Functions

A boundary Nevanlinna-Pick interpolation problem is posed and solved in the quaternionic setting. Given nonnegative real numbers k(1),...,k(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) not equal 1 for u = 1,..., N, and quaternions s(1),...,s(N), we wish to find a slice hyperholomorphic Schur function s so that lim(r -> 1r is an element of(0,1)) s(rp(u)) = s(u) for u = 1,...,N, and lim(r -> 1r is an element of(0,1)) 1-s(rp(u))(s(u)) over bar /1-r <= k(u), for u = 1,...,N. Our arguments rely on the theory of slice hyperholomorphic functions and reproducing kernel Hilbert spaces.