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we start the study of Schur analysis in the quaternionic setting using thetheory of slice hyperholomorphic functions. The novelty of our approach is thatslice hyperholomorphic functions allows to write realizations in terms of asuitable resolvent, the so called S-resolvent operator and to extend severalresults that hold in the complex case to the quaternionic case. We discussreproducing kernels, positive definite functions in this setting and we showhow they can be obtained in our setting using the extension operator and theslice regular product. We define Schur multipliers, and find their co-isometricrealization in terms of the associated de Branges-Rovnyak space.

Schur functions and their realizations in the slice hyperholomorphic setting