Complex Analysis and Operator Theory

, Volume 13, Issue 5, pp 2069–2093 | Cite as

Pick Interpolation on the Polydisc: Small Families of Sufficient Kernels

  • Gautam BharaliEmail author
  • Vikramjeet Singh Chandel


We address Pick’s interpolation problem on the unit polydisc in \(\mathbb {C}^n\), \(n\ge 2\), by characterizing all interpolation data that admit a \(\mathbb {D}\)-valued interpolant in terms of a family of positive-definite kernels parametrized by a class of polynomials. This uses a duality approach that has been associated with Pick interpolation, together with some approximation theory. Furthermore, we use duality methods to understand the set of points on the n-torus at which the boundary values of a given solution to an extremal interpolation problem are not unimodular.


Dual algebra Kernels Pick–Nevanlinna interpolation Polydisc Weak-star topology 

Mathematics Subject Classification

Primary 32A25 46E20 Secondary 32A38 46J15 


  1. 1.
    Agler, J.: Interpolation, unpublished manuscript (1988)Google Scholar
  2. 2.
    Agler, J., McCarthy, J.E.: Nevanlinna–Pick interpolation on the bidisk. J. Reine Angew. Math. 506, 191–204 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Agler, J., McCarthy, J.E.: Pick Interpolation and Hilbert Function Spaces. American Mathematical Society, Providence (2002)CrossRefzbMATHGoogle Scholar
  4. 4.
    Amar, E., Thomas, P.J.: Finite interpolation with minimum uniform norm in \({\mathbb{C}}^{n}\). J. Funct. Anal. 170(2), 512–525 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ball, J.A., Trent, T.T.: Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna–Pick interpolation in several variables. J. Funct. Anal. 157(1), 1–61 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bercovici, H., Westwood, D.: The factorization of functions in the polydisc. Houst. J. Math. 18(1), 1–6 (1992)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cole, B., Lewis, K., Wermer, J.: Pick conditions on a uniform algebra and von Neumann inequalities. J. Funct. Anal. 107(2), 235–254 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cole, B.J., Wermer, J.: Pick interpolation, von Neumann inequalities, and hyperconvex sets. In: Gauthier, P.M., Sabidussi, G. (eds) Complex Potential Theory, pp. 89–129. Kluwer Academic Publishers, Dordrecht, (1994)Google Scholar
  9. 9.
    Conway, J.B.: A Course in Functional Analysis. Springer, New York (1990)zbMATHGoogle Scholar
  10. 10.
    Conway, J.B.: A Course in Operator Theory, Graduate Studies in Mathematics. American Mathematical Society, Providence (1999)Google Scholar
  11. 11.
    Davidson, K.R., Hamilton, R.: Nevanlinna–Pick interpolation and factorization of linear functionals. Integral Equ. Oper. Theory 70(1), 125–149 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hamilton, R.: Pick interpolation and the distance formula, Ph.D. thesis (2012),
  13. 13.
    Hamilton, R.: Pick interpolation in several variables. Proc. Am. Math. Soc. 141(6), 2097–2103 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kosiński, L.: Three-point Nevanlinna–Pick problem in the polydisc. Proc. Lond. Math. Soc. 111(4), 887–910 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    McCullough, S.: Nevanlinna–Pick type interpolation in a dual algebra. J. Funct. Anal. 135(1), 93–131 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Rudin, W.: Function Theory in Polydiscs. W.A. Benjamin, New York (1969)zbMATHGoogle Scholar
  17. 17.
    Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)zbMATHGoogle Scholar
  18. 18.
    Sarason, D.: Generalized interpolation in \(H^{\infty }\). Trans. Am. Math. Soc. 127, 179–203 (1967)MathSciNetzbMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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