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About a Non-Standard Interpolation Problem

  • Daniel Alpay
  • Alain Yger
Article
  • 28 Downloads

Abstract

Using algebraic methods, and motivated by the one variable case, we study a multipoint interpolation problem in the setting of several complex variables. The duality realized by the residue generator associated with an underlying Gorenstein algebra, using the Lagrange interpolation polynomial, plays a key role in the arguments.

Keywords

Residue theory Interpolation 

Mathematics Subject Classification

Primary 32A27 Secondary 13P15 

References

  1. 1.
    Agler, J.: On the representation of certain holomorphic functions defined on a polydisk, volume 48 of Operator Theory: advances and applications, pp. 47–66. Birkhäuser, Basel (1990)Google Scholar
  2. 2.
    Aizenberg, L.A., Yuzhakov, A.P.: Integral Representations and Residues in Multidimensional Complex Analysis, volume 58 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI, 1983. Translated from the 1979 Russian original by H. H. McFaden, Translation edited by Lev J. LeifmanGoogle Scholar
  3. 3.
    Alpay, D., Jorgensen, P., Lewkowicz, I., Volok, D.: A new realization of rational functions, with applications to linear combination interpolation, the Cuntz relations and kernel decompositions. Complex Var. Elliptic Equ. 61(1), 42–54 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Alpay, D., Kaptanoğlu, H.T.: Some finite-dimensional backward shift-invariant subspaces in the ball and a related interpolation problem. Integral Equ. Oper. Theory 42, 1–21 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ball, J.A., Bolotnikov, V.: Realization and interpolation for Schur–Agler-class functions on domains with matrix polynomial defining function in \({\mathbb{C}}^n\). J. Funct. Anal. 213(1), 45–87 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ball, J.A., Gohberg, I., Rodman, L.: Interpolation of rational matrix functions, volume 45 of operator theory: advances and applications. Birkhäuser, Basel (1990)CrossRefGoogle Scholar
  7. 7.
    Ball, J.A., Kaliuzhnyi-Verbovetskyi, D.S.: Schur–Agler and Herglotz–Agler classes of functions: positive-kernel decompositions and transfer-function realizations. Adv. Math. 280, 121–187 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Berenstein, C.A., Gay, R., Vidras, A., Yger, A.: Residue currents and Bezout identities, volume 114 of progress in mathematics. Birkhäuser, Basel (1993)CrossRefGoogle Scholar
  9. 9.
    Berenstein, C.A., Yger, A.: Residue calculus and effective Nullstellensatz. Am. J. Math. 121(4), 723–796 (1999)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Boyer, J.-Y., Hickel, M.: Extension dans un cadre algébrique d’une formule de Weil. Manuscr. Math. 98(2), 195–223 (1999)CrossRefGoogle Scholar
  11. 11.
    Cardinal, J.P., Mourrain, B.: Algebraic approach of residues and Applications. In: The Mathematics of Numerical Analysis (Proc. 1995 AMS-SIAM Summer Seminar in Applied Mathematics, Park City, UT, 1995, J. Renegar, M. Shub, S. Smale eds.), volume 32 of Lectures in Appl. Math., pp. 189–210. American Mathematical Society, Providence, RI (1996)Google Scholar
  12. 12.
    de Boor, C.: Ideal interpolation. In: Chui, C.K., Neamtu, M., Schumaker, L.L. (eds.) Approximation Theory XI. Gaitlinburg 2004, pp. 59–91. Nashboro Press, Brentwood, TR (2005)Google Scholar
  13. 13.
    Elkadi, M., Mourrain, B.: Introduction à la résolution des systèmes polynomiaux, volume 59 of Mathématiques & Applications (Berlin). Springer, Berlin (2007)CrossRefGoogle Scholar
  14. 14.
    Gleason, A.: The Cauchy-Weil theorem. J. Math. Mech. 12, 429–444 (1963)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley-Intersciences, New York (1978)zbMATHGoogle Scholar
  16. 16.
    Jelonek, Z.: On the effective Nullstellensatz. Invent. Math. 162(1), 1–17 (2005)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Kollár, J.: Sharp effective nullstellensatz. J. Am. Math. Soc. 1(4), 963–975 (1988)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Płoski, A.: On the Noether exponent. Bull. Soc. Sci. Lett. Lédz 40, Sér. Rech. Déform. 8, 23–29 (1990)Google Scholar
  19. 19.
    Samuelsson, H.: Analytic continuation of residue currents. Ark. Mat. 47(1), 127–141 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Tsikh, A.K.:Multidimensional Residues and Their Applications, volume 103 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (1992). Translated from the 1988 Russian original by E. J. F. Primrose, Translation edited by S. GelfandGoogle Scholar
  21. 21.
    Tsikh, A., Yger, A.: Residue currents. J. Math. Sci. (N. Y.) 120(6), 1916–1971 (2004). (Complex analysis)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Weil, A.: L’intégrale de Cauchy et les fonctions de plusieurs variables. Math. Ann. 111, 178–182 (1935)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Schmid College of Science and TechnologyChapman UniversityOrangeUSA
  2. 2.Institut de MathématiquesUniversité de BordeauxTalenceFrance

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