Skip to main content
SpringerLink
Log in

On tangential interpolation in reproducing kernel Hilbert modules and applications

  • Chapter
Topics in Interpolation Theory

Part of the book series: Operator Theory Advances and Applications ((OT,volume 95))

Abstract

Nevanlinna-Pick and Carathéodory-Fejér interpolation problems are considered in reproducing kernel Hilbert modules. Applications are given to interpolation in Hardy-Sobolev, Dirichlet and Bergman spaces. The case of de Branges-Rovnyak spaces is also considered.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Agler. Nevanlinna-Pick interpolation on Sobolev spaces. Proc. Amer. Math. Soc., 108: 341–351, 1986.

    Article  MathSciNet  Google Scholar 

  2. D. Alpay and V. Bolotnikov. Two sided Nevanlinna-Pick interpolation for a class of matrix-valued functions. Zeitschrift für Analysis und ihre Anwendungen, 12: 211–238, 1993.

    MathSciNet  MATH  Google Scholar 

  3. D. Alpay and V. Bolotnikov. The Carathéodory-Fejér H2 interpolation problem. Zeitschrift für Analysis und ihrer Anwendungen, 13: 583–597, 1994.

    MathSciNet  MATH  Google Scholar 

  4. D. Alpay and V. Bolotnikov. Two sided interpolation for matrix functions with entries in the Hardy space. Lin. Alg. Appl., 223 /224: 31–56, 1995.

    Article  MathSciNet  Google Scholar 

  5. D. Alpay, V. Bolotnikov, and Ph. Loubaton. On tangential H2 interpolation with second order norm constraints. Integral Equations Operator Theory, 24: 156–178, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Alpay, V. Bolotnikov, and Y. Peretz. On the tangential interpolation problem for H2 functions. Transactions of the American Mathematical Society, 347: 675–686, 1995.

    MathSciNet  MATH  Google Scholar 

  7. D. Alpay, P. Bruinsma, A. Dijksma, and H.S.V. de Snoo. Interpolation problems, extensions of symmetric operators and reproducing kernel spaces II. Integral Equations Operator Theory, 14: 465–500, 1991.

    Google Scholar 

  8. D. Alpay, P. Dewilde, and H. Dym. On the existence and construction of solutions to the partial lossless inverse scattering problem with applications to estimation theory. IEEE Transactions on Information theory, 35: 1184–1205, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Alpay, P. Dewilde, and H. Dym. Lossless inverse scattering and reproducing kernels for upper triangular operators, volume 47 of Operator Theory: Advances and Applications, pages 61–133. Birkhäuser Verlag, Basel, 1990.

    Google Scholar 

  10. D. Alpay, A. Dijksma, H. de Snoo, and J. Rovnyak. Schur functions, operator colligations, and reproducing kernel Pontryagin spaces. Preprint, 1995.

    Google Scholar 

  11. D. Alpay and H. Dym. Hilbert spaces of analytic functions, inverse scattering and operator models, I. Integral Equations Operator Theory, 7: 589–641, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  12. N. Aronszajn. Theory of reproducing kernels. Transactions of the American Mathematical Society, 68: 337–404, 1950.

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Arov. The generalized bitangential Carathéodory-Nevanlinna-Pick problem and (j, J0)-inner matrix-functions. Russian Acad. Sci. Izv. Math., 42: 1–26, 1994.

    MathSciNet  Google Scholar 

  14. T.Ya. Azizov and I.S. Iohvidov. Foundations of the theory of linear operators in spaces with indefinite metric. Nauka, Moscow, 1986. (Russian). English translation: Linear operators in spaces with an indefinite metric. John Wiley, New-York, 1989.

    Google Scholar 

  15. J. Ball and J.W. Helton. A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation theory. Journal of Operator Theory, 8: 107–142, 1983.

    MathSciNet  Google Scholar 

  16. L. Baratchart Interpolation and Fourier coefficients in the Hardy space H2. In M. Kaashoek, J.H. van Schuppen, and A.C.M. Ran, editors, Proceedings of the international symposium MTNS 89, volume 3 of Progress in systems and control theory, pages 387–394. Birkhäuser Verlag, Basel, 1990.

    Google Scholar 

  17. L. Baratchart and M. Zerner. On the recovery of functions from pointwise boundary values in a Hardy-Sobolev space of the disk. Journal of Computational and Applied Mathematics, 46: 255–269, 1993.

    Article  MathSciNet  Google Scholar 

  18. S. Bergman and M. Schiffer. Kernel functions and elliptic differential equations in mathematical physics. Academic Press, 1953.

    Google Scholar 

  19. L. de Branges. Espaces hilbertiens de fonctions entières. Masson, Paris, 1972.

    Google Scholar 

  20. L. de Branges and J. Rovnyak. Canonical models in quantum scattering theory. In C. Wilcox, editor, Perturbation theory and its applications in quantum mechanics. Holt, Rinehart and Winston, New-York, 1966.

    Google Scholar 

  21. L. de Branges and J. Rovnyak. Square summable power series. Holt, Rinehart and Winston, New-York, 1966.

    Google Scholar 

  22. B. Davis and J. McMarthy. Multipliers in de Branges spaces. Michigan Math. J., 38: 225–240, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Dritschel and J. Rovnyak. Extensions theorems for contractions on Kreĭn spaces, volume 47 of Operator theory: Advances and Applications, pages 221–305. Birkhäuser Verlag, Basel, 1990.

    Google Scholar 

  24. P.L. Duren and D.L. Williams. Interpolation problems in function spaces. Journal of Functional Analysis, 9: 75–86, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Dym. J contractive matrix functions, reproducing kernel spaces and interpolation, volume 71 of CBMS Lecture Notes. Amer. Math. Soc., Rhodes Island, 1989.

    Google Scholar 

  26. P.A. Fuhrmann. Linear systems and operators in Hilbert space. McGraw-Hill international book company, 1981.

    Google Scholar 

  27. I. Gohberg, editor. Time variant systems and interpolation, volume 56 of Operator theory: Advances and Applications. Birkhäuser Verlag, Basel, 1992.

    Google Scholar 

  28. J. Helton. Operator theory, analytic functions, matrices and electrical engineering, volume 68 of CBMS Lecture Notes. Amer. Math. Soc., Rhodes Island, 1987.

    Google Scholar 

  29. J.W. Helton and J. Ball. The cascade decomposition of a given system vs the linear fractional decompositions of its transfer function. Integral Equations and Operator Theory, 5: 314–385, 1982.

    Google Scholar 

  30. S. Itoh. Reproducing kernels in modules over C*-algebras and their applications. Bull. Kyushu Inst. Tech., 37: 1–20, 1990.

    MATH  Google Scholar 

  31. A. Kheifets. Generalized bitangential Schur-Nevanlinna-Pick problem, related Parseval equality and scattering operator. deposited in VINITI, 11.05. 1989. No. 3108 B89 Dep., 1–60, 1989, russian.

    Google Scholar 

  32. A. Kheifets. Parseval equality in abstract interpolation problem and coupling of open systems. Jour. Sov. Math., 49: 1307–1310, 1990.

    Article  MathSciNet  Google Scholar 

  33. M.G. Kreĭn and A.A. Nudelman. The Markov moment problem and extremal problems, volume 50 of Translations of mathematical monographs. American Mathematical Society, Providence, Rhode Island, 1977.

    Google Scholar 

  34. P.D. Lax and R.S. Phillips. Scattering theory. Academic Press, New-York, 1967.

    Google Scholar 

  35. B. Lotto and D. Sarason. Multipliers in de Branges-Rovnyak spaces. Indiana University Mathematical Journal, 42: 907–920, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  36. W. Paschke. Inner product spaces over B*-algebras. Transactions of the American Mathematical Society, 1982: 443–468, 1973.

    MathSciNet  Google Scholar 

  37. M. Rosenblum and J. Rovnyak. Hardy classes and operator theory. Oxford University Press, 1985.

    Google Scholar 

  38. W. Rudin. Real and complex analysis. Mc Graw Hill, 1982.

    Google Scholar 

  39. S. Saitoh. Theory of reproducing kernels and its applications, volume 189. Longman scientific and technical, 1988.

    Google Scholar 

  40. D. Sarason. Exposed points in H1, I, volume 41 of Operator Theory: Advances and Applications, pages 485–496. Birkhäuser Verlag, Basel, 1989.

    Google Scholar 

  41. L. Schwartz. Sous espaces hilbertiens d’espaces vectoriels topologiques et noyaux associés (noyaux reproduisants). J. d’Analyse Math., 13: 115–256, 1964.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Daniel Alpay

  2. Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot, 76100, Israel

    Vladimir Bolotnikov

Authors
  1. Daniel Alpay
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir Bolotnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Theoretical Mathematics, The Weizmann Institute of Science, 76100, Rehovot, Israel

    H. Dym  & V. Katsnelson  & 

  2. Mathematisches Institut, Universität Leipzig, 04109, Leipzig, Germany

    B. Fritzsche  & B. Kirstein  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer Basel AG

About this chapter

Cite this chapter

Alpay, D., Bolotnikov, V. (1997). On tangential interpolation in reproducing kernel Hilbert modules and applications. In: Dym, H., Katsnelson, V., Fritzsche, B., Kirstein, B. (eds) Topics in Interpolation Theory. Operator Theory Advances and Applications, vol 95. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8944-5_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8944-5_3

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9838-6

  • Online ISBN: 978-3-0348-8944-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation