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Interpolation Problems for Certain Classes of Slice Hyperholomorphic Functions

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Abstract

A general interpolation problem (which includes as particular cases the Nevanlinna–Pick and Carathéodory–Fejér interpolation problems) is considered in two classes of slice hyperholomorphic functions of the unit ball of the quaternions. In the Hardy space of the unit ball we present a Beurling–Lax type parametrization of all solutions, and the formula for the minimal norm solution. In the class of functions slice hyperholomorphic in the unit ball and bounded by one in modulus there (that is, in the class of Schur functions in the present framework) we present a necessary and sufficient condition for the problem to have a solution, and describe the set of all solutions in the indeterminate case.

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Author notes

  1. Daniel Alpay

    Present address: Department of Mathematics, Chapman University, One University Drive, Orange, CA, 92866, USA

Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, 84105, Israel

    Daniel Alpay

  2. Department of Mathematics, The College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Vladimir Bolotnikov

  3. Dipartimento di Matematica, Politecnico di Milano, Via E. Bonardi, 9, 20133, Milan, Italy

    Fabrizio Colombo & Irene Sabadini

Authors
  1. Daniel Alpay
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  2. Vladimir Bolotnikov
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  3. Fabrizio Colombo
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  4. Irene Sabadini
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Corresponding author

Correspondence to Vladimir Bolotnikov.

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Alpay, D., Bolotnikov, V., Colombo, F. et al. Interpolation Problems for Certain Classes of Slice Hyperholomorphic Functions. Integr. Equ. Oper. Theory 86, 165–183 (2016). https://doi.org/10.1007/s00020-016-2318-x

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  • DOI: https://doi.org/10.1007/s00020-016-2318-x

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