Reproducing Kernel Spaces and Applications

1. Front Matter

   Pages i-xv

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2. Realization of Functions into the Symmetrised Bidisc

   • J. Agler, F. B. Yeh, N. J. Young

   Pages 1-37

3. A Basic Interpolation Problem for Generalized Schur Functions and Coisometric Realizations

   • D. Alpay, T. Ya. Azizov, A. Dijksma, H. Langer, G. Wanjala

   Pages 39-76

4. Formal Reproducing Kernel Hilbert Spaces: The Commutative and Noncommutative Settings

   • Joseph A. Ball, Victor Vinnikov

   Pages 77-134

5. On Realizations of Rational Matrix Functions of Several Complex Variables II

   • M. F. Bessmertnyĭ

   Pages 135-146

6. Bergman Projection and Weighted Holomorphic Functions

   • Der-Chen Chang, Robert Gilbert, Jingzhi Tie

   Pages 147-169

7. Linear Fractional Transformations, Riccati Equations and Bitangential Interpolation, Revisited

   • Harry Dym

   Pages 171-212

8. A Class of Matrix-valued Schrödinger Operators with Prescribed Finite-band Spectra

   • Fritz Gesztesy, Lev A. Sakhnovich

   Pages 213-253

9. Laplace Transform Asymptotics, Bergman Kernels and Composition Operators

   • Thomas L. Kriete

   Pages 255-272

10. On the Structure of Self-similar Systems: A Hilbert Space Approach

    • Mamadou Mboup

    Pages 273-302

11. Reproducing Kernels and a Family of Bounded Linear Operators

    • Saburou Saitoh

    Pages 303-312

12. Multipliers in the Reproducing Kernel Hilbert Space, Subnormality and Noncommutative Complex Analysis

    • Franciszek Hugon Szafraniec

    Pages 313-331

13. Existence of Unitary Dilations as a Moment Problem

    • F.-H. Vasilescu

    Pages 333-344

20. Pattern recognition and statistical learning theory (the theory of support vector machines). See [40], [58]. In this last volume we refer in particular to the papers [63] and [64]. Since this topic is maybe less known to the operator theory community we mention that the support vector method is a general approach to function estimation problems. See [63, p. 26]. We note that the above list and the given references are by no way exhaustive. We refer to the first section of the paper of S. Saitoh in the present volume for another (and mainly different) list of topics where reproducing kernel spaces appear. Quite often a given question is best understood in a reproducing kernel Hilbert space (for instance when using Cauchy's formula in the Hardy space H ) 2 and one finds oneself as Mr Jourdain of Moliere' Bourgeois Gentilhomme speaking Prose without knowing it [48, p. 51]: Par ma foil il y a plus de quarante ans que je dis de la prose sans que l j'en susse rien.

• Complex analysis
• Functional analysis
• Hilbert space
• Operator theory
• PDEs
• partial differential equations

• Department of Mathematics, Ben Gurion University of the Negev, Beer Sheva, Israel

  Daniel Alpay

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