Reproducing kernel kreĭn spaces

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(1)Daniel Alpay Editor. Operator Theory. 1 3Reference.

(2) Operator Theory.

(3)

(4) Daniel Alpay Editor. Operator Theory With 51 Figures and 2 Tables.

(5) Editor Daniel Alpay Earl Katz Chair in Algebraic System Theory Department of Mathematics Ben-Gurion University of the Negev Be’er Sheva, Israel. ISBN 978-3-0348-0666-4 ISBN 978-3-0348-0667-1 (eBook) ISBN 978-3-0348-0668-8 (print and electronic bundle) DOI 10.1007/978-3-0348-0667-1 Library of Congress Control Number: 2015939608 Mathematics Subject Classification: (main) 47-00, (secondary) 46-00, 93-00, 30-00 Springer Basel Heidelberg New York Dordrecht London © Springer Basel 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media (www.springer.com).

(6) Preface. Welcome to Operator Theory. A one-sentence definition of operator theory could be: The study of (linear) continuous operations between topological vector spaces, these being in general (but not exclusively) Fréchet, Banach, or Hilbert spaces (or their duals). Operator theory is thus a very wide field, with numerous facets and a host of applications, both applied and theoretical. There are deep connections with complex analysis, functional analysis, mathematical physics, and electrical engineering, to name a few. Fascinating new applications and directions regularly appear, such as operator spaces, free probability, and Clifford analysis. The present handbook contains a collection of reference papers, which survey various aspects of modern operator theory. In our choice of the sections, we tried to reflect the underlying diversity, and any choice of topics necessarily leaves away other aspects. This is a dynamic ongoing project, and more sections are planned in the online version to complete the picture. We hope you enjoy the reading and profit from this endeavor. The work is divided into eight sections. There is no single path to present operator theory in its various facets, and we have chosen the following way: The opening section, edited by Franciszek Hugon Szafraniec, is devoted to reproducing kernel spaces. Next, indefinite inner product spaces are considered in the section edited by Matthias Langer and Harald Woracek. The third section, edited by Anton Baranov and Harald Woracek, discusses an important class of reproducing kernel spaces, namely de Branges Rovnyak spaces. There is a continuous feedback loop between linear system theory and signal processing on the one hand, and operator theory on the other hand. Some of these aspects are presented in the section edited by Mamadou Mboup and the editor. Modern operator theory goes far beyond the one complex variable setting, and the last four sections discuss some of these aspects. Multivariable operator theory is considered in Section 5, edited by Joseph A. Ball. Some aspects of infinite dimensional analysis are studied in Section 6, edited by Palle E.T. Jorgensen. Last but not least, the last two sections, edited by Fabrizio Colombo, Irene Sabadini, and Michael Shapiro, consider a fascinating noncommutative version of operator and function theory, when the complex numbers are replaced by the field of quaternions or by a Clifford algebra. v.

(7) vi. Preface. It is a pleasure to thank the various section editors for their dedicated work, the referees, and the Springer team (in particular Ms. Audrey Wong). Last but not least, special thanks are due to Dr. Thomas Hempfling for his support and help all along the project. Be’er Sheva, Israel. Daniel Alpay.

(8) Contents. Volume 1 Part I Reproducing Kernel Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . Franciszek Hugon Szafraniec 1. 2. 3. 1. The Reproducing Kernel Property and Its Space: The Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Franciszek Hugon Szafraniec. 3. The Reproducing Kernel Property and Its Space: More or Less Standard Examples of Applications . . . . . . . . . . . . . . . . Franciszek Hugon Szafraniec. 31. The Use of Kernel Functions in Solving the Pick Interpolation Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jim Agler and John E. McCarthy. 59. 4. Bergman Kernel in Complex Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . Łukasz Kosiński and Włodzimierz Zwonek. 73. 5. Sampling Theory and Reproducing Kernel Hilbert Spaces . . . . . . . . Antonio G. García. 87. 6. Reproducing Kernels in Coherent States, Wavelets, and Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Syed Twareque Ali. 7. Geometric Perspectives on Reproducing Kernels . . . . . . . . . . . . . . . . . 127 Daniel Belti¸taˇ and José E. Galé. Part II Indefinite Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . Matthias Langer and Harald Woracek 8. 149. Multi-valued Operators/Linear Relations Between Kre˘ın Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Hendrik Luit Wietsma vii.

(9) viii. 9. Contents. Symmetric and Isometric Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Hendrik Luit Wietsma. 10 Boundary Triplets, Weyl Functions, and the Kre˘ın Formula . . . . . . . 183 Vladimir Derkach 11 Contractions and the Commutant Lifting Theorem in Kre˘ın Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 Michael Dritschel 12 Locally Definitizable Operators: The Local Structure of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 Carsten Trunk 13 Schur Analysis in an Indefinite Setting . . . . . . . . . . . . . . . . . . . . . . . . . 261 Aad Dijksma 14 Reproducing Kernel Kre˘ın Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Aurelian Gheondea 15 Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Annemarie Luger 16 Indefinite Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 Michael Kaltenbäck 17 The Critical Point Infinity Associated with Indefinite Sturm–Liouville Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 Andreas Fleige 18 Finite-Dimensional Indefinite Inner Product Spaces and Applications in Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . 431 Christian Mehl 19 The Algebraic Riccati Equation and Its Role in Indefinite Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 André C.M. Ran Part III de Branges Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anton Baranov and Harald Woracek. 471. 20 Hilbert Spaces of Entire Functions: Early History . . . . . . . . . . . . . . . . 473 James Rovnyak 21 de Branges Spaces and Growth Aspects . . . . . . . . . . . . . . . . . . . . . . . . . 489 Harald Woracek 22 Two-Dimensional Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . . . . . 525 Henrik Winkler.

(10) Contents. ix. 23 de Branges Spaces and Kre˘ın’s Theory of Entire Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 549 Luis O. Silva and Julio H. Toloza 24 The Beurling–Malliavin Multiplier Theorem and Its Analogs for the de Branges Spaces . . . . . . . . . . . . . . . . . . . . . . 581 Yurii Belov and Victor Havin 25 Jacobi Matrices and de Branges Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 609 Roman Romanov 26 Schrödinger Operators and Canonical Systems . . . . . . . . . . . . . . . . . . 623 Christian Remling 27 de Branges–Rovnyak Spaces: Basics and Theory . . . . . . . . . . . . . . . . . 631 Joseph A. Ball and Vladimir Bolotnikov 28 de Branges–Rovnyak Spaces and Norm-Constrained Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681 Joseph A. Ball and Vladimir Bolotnikov 29 de Branges Spaces of Vector-Valued Functions . . . . . . . . . . . . . . . . . . . 721 Damir Z. Arov and Harry Dym 30 Applications of de Branges Spaces of Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753 Damir Z. Arov and Harry Dym Part IV Linear Systems Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel Alpay and Mamadou Mboup. 777. 31 Realization of Herglotz–Nevanlinna Functions by Conservative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779 Yury Arlinski˘ı, Sergey Belyi, and Eduard Tsekanovski˘ı 32 Synchronization Problems for Spatially Invariant Infinite Dimensional Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 811 Avraham Feintuch 33 Linear Transformations in Signal and Optical Systems . . . . . . . . . . . 833 Ahmed I. Zayed 34 Perturbations of Unbounded Fredholm Linear Operators in Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875 Toka Diagana 35 Robust Stabilization of Linear Control Systems Using a Frequency Domain Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 881 Amol Sasane.

(11) x. Contents. 36 Semi- and Quasi-separable Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 901 Patrick Dewilde and Alle-Jan Van der Veen 37 Basics of Secrecy Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931 Phillip A. Regalia. Volume 2 Part V Multivariable Operator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . Joseph A. Ball. 967. 38 An Introduction to Hilbert Module Approach to Multivariable Operator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969 Jaydeb Sarkar 39 Applications of Hilbert Module Approach to Multivariable Operator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035 Jaydeb Sarkar 40 Commutative Dilation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1093 Caline Ambrozie and Vladimír Müller 41 Operator Theory and Function Theory in Drury–Arveson Space and Its Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125 Orr Shalit 42 Taylor Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1181 Vladimír Müller Part VI Infinite Dimensional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217 Palle E.T. Jorgensen 43 Unbounded Operators, Lie Algebras, and Local Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1221 Palle E.T. Jorgensen and Feng Tian 44 Arithmetic Functions in Harmonic Analysis and Operator Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245 Ilwoo Cho and Palle E.T. Jorgensen 45 A Von Neumann Algebra over the Adele Ring and the Euler Totient Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285 Ilwoo Cho and Palle E.T. Jorgensen Part VII General Aspects of Quaternionic and Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1337 Fabrizio Colombo, Irene Sabadini, and Michael Shapiro 46 Introductory Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1339 Frank Sommen and Hennie De Schepper.

(12) Contents. xi. 47 Quaternionic Analysis: Application to Boundary Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1369 Klaus Gürlebeck and Wolfgang Sprößig 48 Function Spaces in Quaternionic and Clifford Analysis . . . . . . . . . . . 1393 Mircea Martin 49 Quaternionic Analysis and Some Conventional Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1423 Michael Shapiro 50 Quaternionic and Clifford Analysis for Non-smooth Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447 Ricardo Abreu-Blaya and Juan Bory-Reyes 51 Clifford Analysis for Higher Spin Operators . . . . . . . . . . . . . . . . . . . . 1471 David Eelbode 52 Fueter Mapping Theorem in Hypercomplex Analysis . . . . . . . . . . . . . 1491 Tao Qian 53 Representation Theory in Clifford Analysis . . . . . . . . . . . . . . . . . . . . . 1509 Vladimír Souˇcek 54 Quaternionic and Clifford Analysis in Several Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1549 Daniele C. Struppa Part VIII Further Developments of Quaternionic and Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1579 Fabrizio Colombo, Irene Sabadini, and Michael Shapiro 55 Hermitian Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1581 Irene Sabadini and Frank Sommen 56 Discrete Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1609 Uwe Kähler and Frank Sommen 57 Slice Hyperholomorphic Functions with Values in Some Real Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1631 Daniele C. Struppa 58 Fourier Transforms in Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . 1651 Hendrik De Bie 59 Wavelets in Clifford Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1673 Swanhild Bernstein 60 Monogenic Signal Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1701 Paula Cerejeiras and Uwe Kähler 61 Some Peculiarities of Quaternionic Linear Spaces . . . . . . . . . . . . . . . . 1725 Maria Elena Luna-Elizarrarás and Michael Shapiro.

(13) xii. Contents. 62 Schur Analysis in the Quaternionic Setting: The Fueter Regular and the Slice Regular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745 Daniel Alpay, Fabrizio Colombo, and Irene Sabadini 63 Slice Hyperholomorphic Functional Calculi . . . . . . . . . . . . . . . . . . . . . 1787 Fabrizio Colombo and Irene Sabadini 64 The Monogenic Functional Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . 1823 Brian Jefferies Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1853.

(14) Editorial Board. Editor Daniel Alpay Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel Section Editors Part I: Reproducing Kernel Hilbert Spaces Franciszek Hugon Szafraniec Instytut Matematyki, Uniwersytet Jagiellónski, Kraków, Poland Part II: Indefinite Inner Product Spaces Matthias Langer Department of Mathematics and Statistics, University of Strathclyde, Glasgow, Scotland, UK Harald Woracek Institut for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria Part III: de Branges Spaces Anton Baranov Department of Mathematics and Mechanics, St. Petersburg State University, Pedrodvorets, Russia Harald Woracek Institut for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria Part IV: Linear Systems Theory Daniel Alpay Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel Mamadou Mboup Université de Reims Champagne Ardenne, CReSTIC - UFR des Sciences Exactes et Naturelles Moulin de la Housse, Reims, France Part V: Multivariable Operator Theory Joseph A. Ball Department of Mathematics, Virginia Tech, Blacksburg, VA, USA. xiii.

(15) xiv. Editorial Board. Part VI: Infinite Dimensional Analysis Palle E.T. Jorgensen Department of Mathematics, The University of Iowa, Iowa City, IA, USA Parts VII and VIII: General Aspects of Quaternionic and Clifford Analysis and Further Developments of Quaternionic and Clifford Analysis Fabrizio Colombo Dipartimento di Matematica, Politecnico di Milano, Milano, Italy Irene Sabadini Dipartimento di Matematica, Politecnico di Milano, Milano, Italy Michael Shapiro Departamento de Matemáticas, Escuela Superior de Física y Matemáticas, del Instituto Politécnico Nacional, Mexico City, Mexico.

(16) Contributors. Ricardo Abreu-Blaya Faculty of Mathematics and Informatics, University of Holguin, Holguin, Cuba Jim Agler University of California, San Diego, La Jolla, CA, USA Syed Twareque Ali Department of Mathematics and Statistics, Concordia University, Montréal, QC, Canada Daniel Alpay Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel Caline Ambrozie Institute of Mathematics, Academy of Czech Republic, Prague, Czech Republic Yury Arlinski˘ı Department of Mathematics, East Ukrainian National University, Lugansk, Ukraine Damir Z. Arov South Ukrainian National Pedagogical University, Odessa, Ukraine Joseph A. Ball Department of Mathematics, Virginia Tech, Blacksburg, VA, USA Yurii Belov Chebyshev Laboratory, St. Petersburg State University, Vasilyevsky Island, St. Petersburg, Russia Daniel Belti¸taˇ Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest, Romania Sergey Belyi Department of Mathematics, Troy University, Troy, AL, USA Swanhild Bernstein Institut für Angewandte Analysis, Fakultät für Mathematik und Informatik, Technische Universität Bergakademie Freiberg, Freiberg, Germany Vladimir Bolotnikov Department of Mathematics, The College of William and Mary, Williamsburg, VA, USA Juan Bory-Reyes Department of Mathematics, Universidad de Oriente, Santiago de Cuba, Cuba. xv.

(17) xvi. Contributors. Paula Cerejeiras Department of Mathematics, CIDMA – Center for Research and Development in Mathematics and Applications, University of Aveiro, Aveiro, Portugal Ilwoo Cho Department of Mathematics, St. Ambrose University, Davenport, IA, USA Fabrizio Colombo Dipartimento di Matematica, Politecnico di Milano, Milano, Italy Hendrik De Bie Department of Mathematical Analysis, Faculty of Engineering and Architecture, Ghent University, Gent, Belgium Vladimir Derkach Department of Mathematics, Donetsk National University, Donetsk, Ukraine Hennie De Schepper Department of Mathematical Analysis, Ghent University, Gent, Belgium Patrick Dewilde Technische Universität München, Institute for Advanced Study, München, Germany Toka Diagana Department of Mathematics, Howard University, College of Arts and Sciences, Washington, DC, USA Aad Dijksma Johann Bernoulli Institute of Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands Michael Dritschel School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne, UK Harry Dym The Weizmann Institute of Science, Rehovot, Israel David Eelbode Department of Mathematics and Computer Science, Universiteit Antwerpen, Antwerp, Belgium Avraham Feintuch Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel Andreas Fleige Dortmund, Germany José E. Galé Universidad de Zaragoza and IUMA, Zaragoza, Spain Antonio G. García Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés-Madrid, Spain Aurelian Gheondea Department of Mathematics, Bilkent University, Ankara, Turkey Institute of Mathematics of the Romanian Academy, Bucharest, Romania Klaus Gürlebeck Bauhaus-Universität Weimar, Weimar, Germany.

(18) Contributors. xvii. Victor Havin Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia Brian Jefferies School of Mathematics, The University of New South Wales, Sydney, NSW, Australia Palle E.T. Jorgensen Department of Mathematics, The University of Iowa, Iowa City, IA, USA Uwe Kähler Department of Mathematics, CIDMA – Center for Research and Development in Mathematics and Applications, University of Aveiro, Aveiro, Portugal Michael Kaltenbäck Institut für Analysis und Scientific Computing, TU Wien, Vienna, Austria ´ Łukasz Kosinski Faculty of Mathematics and Computer Science, Department of Mathematics, Jagiellonian University, Kraków, Poland Annemarie Luger Department of Mathematics, Stockholm University, Stockholm, Sweden Maria Elena Luna-Elizarrarás Departamento de Matemáticas, Escuela Superior de Física y Matemáticas, del Instituto Politécnico Nacional, Mexico City, Mexico Mircea Martin Department of Mathematics, Baker University, Baldwin City, KS, USA John E. McCarthy Department of Mathematics, Washington University, St. Louis, MO, USA Christian Mehl Institut für Mathematik, Technische Universität Berlin, Berlin, Germany Vladimír Müller Mathematical Institute, Academy of Sciences of the Czech Republic, Prague, Czech Republic Tao Qian Faculty of Science and Technology, University of Macau, Macau, China André C.M. Ran Department of Mathematics, FEW, VU University Amsterdam, Amsterdam, The Netherlands Unit for BMI, North-West University, Potchefstroom, South Africa Phillip A. Regalia Directorate for Computer and Information Science and Engineering, National Science Foundation, Arlington, VA, USA Christian Remling Mathematics Department, University of Oklahoma, Norman, OK, USA Roman Romanov Department of Physics, St. Petersburg State University, St. Petersburg, Russia James Rovnyak University of Virginia, Charlottesville, VA, USA.

(19) xviii. Contributors. Irene Sabadini Dipartimento di Matematica, Politecnico di Milano, Milano, Italy Jaydeb Sarkar Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore, India Amol Sasane Department of Mathematics, London School of Economics, London, UK Orr Shalit Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel Department of Mathematics, Technion - Israel Institute of Technology, Haifa, Israel Michael Shapiro Departamento de Matemáticas, Escuela Superior de Física y Matemáticas, del Instituto Politécnico Nacional, Mexico City, Mexico Luis O. Silva Departamento de Física Matemática, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Universidad Nacional Autónoma de México, Mexico DF, Mexico Frank Sommen Department of Mathematical Analysis, Ghent University, Gent, Belgium Vladimír Souˇcek Mathematical Institute, Charles University, Prague, Czech Republic Wolfgang Sprößig Institut für Angewandte Analysis, TU Bergakademie Freiberg, Freiberg, Germany Daniele C. Struppa Schmid College of Science and Technology, Chapman University, Orange, CA, USA Franciszek Hugon Szafraniec Instytut Matematyki, Uniwersytet Jagielloński, Kraków, Poland Feng Tian Department of Mathematics, Wright State University, Dayton, OH, USA Julio H. Toloza Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), and Centro de Investigación en Informática para la Ingeniería, Universidad Tecnológica Nacional – Facultad Regional Córdoba, Maestro M. López s/n, Córdoba, Argentina Carsten Trunk Institut für Mathematik, Technische Universität Ilmenau, Ilmenau, Germany Eduard Tsekanovski˘ı Department of Mathematics, Niagara University, New York, NY, USA Alle-Jan Van der Veen Circuits and Systems Section, Delft University of Technology, Delft, The Netherlands.

(20) Contributors. xix. Hendrik Luit Wietsma Department of Mathematics, Stockholm University, Stockholm, Sweden Henrik Winkler Institute of Mathematics, TU Ilmenau, Ilmenau, Germany Harald Woracek Institut for Analysis and Scientific Computing, Vienna University of Technology,Vienna, Austria Ahmed I. Zayed Department of Mathematical Sciences, DePaul University, Schmitt Academic Center (SAC), Chicago, IL, USA Włodzimierz Zwonek Faculty of Mathematics and Computer Science, Department of Mathematics, Jagiellonian University, Kraków, Poland.

(21) Part I Reproducing Kernel Hilbert Spaces Franciszek Hugon Szafraniec. The reproducing property, originated in 1907, has been for over a century one of the most powerful tools in mathematics. It is typically associated with Hilbert spaces of analytic (holomorphic) functions; the kernel being a subordinate issue. According to the philosophy exposed in the monograph by F. H. Szafraniec, Przestrzenie Hilberta z ja¸drem reprodukuja¸cym, (Reproducing kernel Hilbert spaces, in Polish), Wydawnictwo Uniwersytetu Jagiellońskiego, Kraków, 2004, it is just a bridge which links two objects: the Hilbert space of functions and the kernel. This gives both parts involved equal rights. The aforesaid monograph has been distilled resulting in the following chapters: The Reproducing Kernel Property and Its Space: The Basics, and The Reproducing Kernel Property and Its Space: More or Less Standard Examples of Applications. Referring to the customary application in spaces of holomorphic function, there are two chapters: The Use of Kernel Functions in Solving the Pick Interpolation Problem by Jim Agler and John McCarthy and Bergman Kernel in Complex Analysis by Łukasz Kosiński and Włodzimierz Zwonek. For other applications, the choice has been made in favor of the following chapters: Geometric Perspectives on Reproducing Kernels by Daniel Belti¸ta˘ and José Galé, Sampling Theory and Reproducing Kernel Hilbert Spaces by Antonio García, and Reproducing Kernels in Coherent States, Wavelets, and Quantization by Syed Twareque Ali. Needless to say, this a drop in the ocean, and many interesting possibilities are necessarily left untouched..

(22) 1. The Reproducing Kernel Property and Its Space: The Basics Franciszek Hugon Szafraniec. Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Reproducing Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Reproducing Kernel Couple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Positive-Definite Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Completion of Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hilbert Space of Functions Versus Reproducing Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From H to K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From K to H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Constructions of a Positive-Definite Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operations on Reproducing Kernel Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparisons of Reproducing Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sum and Subtraction of Reproducing Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Product of Reproducing Kernels: Tensor Product of Hilbert Spaces . . . . . . . . . . . . . . . . . . Restrictions of Kernels and Interpolation in a Reproducing Kernel Hilbert Space . . . . . . . An Instructive Example ./ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Limit of Spaces with Reproducing Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite-Dimensional Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplication Operators and Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 4 4 5 7 7 8 9 9 10 11 15 15 16 17 18 20 21 22 24 26 28 29. F.H. Szafraniec () Instytut Matematyki, Uniwersytet Jagielloński, Kraków, Poland e-mail: umszafra@cyf-kr.edu.pl © Springer Basel 2015 D. Alpay (ed.), Operator Theory, DOI 10.1007/978-3-0348-0667-1_65. 3.

(23) 4. F.H. Szafraniec. Abstract. This is the first part of the exposition which appears in this handbook under the common title “The Reproducing Kernel Property and Its Space.”. Introduction The foremost intention of this chapter is to develop the anatomy of the reproducing property and its objects: the kernel and the space. This covers more or less the chapter General Theory of Reproducing Kernel Space of the present author’s monograph Przestrzenie Hilberta z ja¸drem reprodukuja¸cym (Hilbert spaces with reproducing kernel, in Polish), Wydawnictwo Uniwersytetu Jagiellońskiego, Kraków 2004, referred here insistently as [21], except the proofs; for them, the monograph [21] is still the best reference source. From the other chapters of [21] (“Elements of dilation theory” and “Spaces of holomorphic functions”), excerpts are in the accompanied chapter The Reproducing Kernel Property and its Space: More or Less Standard Examples of Applications of this handbook; the last of [21], the fourth, entitled “Fock construction,” is skipped.. The Reproducing Property Given a set X , a function of two variables KW X X ! C is customarily called a kernel on X and the sections Kx W x ! K. ; x/, x 2 X are referred to as kernel functions. A unitary space and an inner product space are synonymous here; it is a linear space, always over the field of complex numbers C, endowed with an inner (scalar) product h ; i; sometimes a subscript is used to indicate the space if a confusion may emerge. If the space becomes complete, it is just a Hilbert space.. The Reproducing Kernel Couple Given two objects: 1o A Hilbert space H of complex valued functions on X ; 2o A kernel K on X . If f .x/ D hf; Kx i;. f 2 H; x 2 X. (1.1). provided all the Kx ’s are in H, .K; H/ is said to be a reproducing kernel couple and the equality (1.1) itself a reproducing kernel property (relative to K and H). Besides the fact that (1.1) implies immediately totality (or, in other words, completeness) of the set fKx W x 2 X g, there are two consequences of fundamental importance:.

(24) 1 The Reproducing Kernel Property and Its Space: The Basics. 5. (A) The evaluation functionals ˚x W H 3 f 7! f .x/ 2 C, x 2 X , are continuous. (B) The kernel K is positive definite (the definition follows). Each of (A) and (B) is an independent starting point in constructing the other partner in the couple if one of them is provided for. This gives the members of a reproducing couple equal rights in the process. Moreover, each of the (A) and (B) determines the other uniquely. This justifies the commonly used terminology reproducing kernel for K and reproducing kernel Hilbert space for H; for the latter, the acronym RKHS is in use.. Positive-Definite Kernels A kernel K will be called positive definite if N X. K.xi ; xj /i N j 0;. x1 ; : : : ; xN 2 X; 1 ; : : : N 2 CI. (1.2). i;j D1. the universal quantifier in the formula above refers also to the length N of sequences in the formula; such a situation will be common in what follows. One thing worth to be pointed out is that positive definiteness of kernel K is equivalent to positive definiteness in the sense of linear algebra of each of the matrices 0. 1 K.x1 ; x1 / K.x1 ; xN / B C :: :: :: B C; : : : @ A. x1 ; : : : ; xN 2 X; N D 1; 2; : : :. K.xN ; x1 / K.xN ; xN / Remark 1. Sometimes another way of looking at the kernel is more convenient. Let F .X / denote linear space of all functions W X ! C, which are equal to zero everywhere except for a finite number of xs. Then (cf. [17]) def. k.; / D. X. K.x; y/.x/.y/;. ; 2 F .X /;. x;y2X. is a Hermitian bilinear form (a term proposed by [4]) on F .X /. Moreover, using the convention that i D .xi /, (1.2) can be written in a shorter way k.; / 0;. 2 F .X /:. For a positive-definite kernel K, one has: • Hermitian symmetry K.x; y/ D K.y; x/;. x; y 2 X I. (1.3).

(25) 6. F.H. Szafraniec. • The Schwarz inequality. j. M;N X. K.xi ; yk /i N k j 2. M X. K.xi ; xj /i N j. i;j D1. i;kD1. N X. K.yk ; yl /k N l ;. k;lD1. x1 ; : : : ; xM ; y1 ; : : : ; yN 2 X; 1 ; : : : M ; 1 ; : : : ; N 2 C: Remark 2. It is trivial to note that if K is a positive-definite kernel, then the kernel .x; y/ 7! K.y; x/ is positive definite as well. Here, there is a foretaste of how powerful positive definiteness is. Proposition 1. If X is a topological space and K is a positive-definite kernel on X , then continuity of K with respect to each variable separately at points of the diagonal f.x; x/W x 2 X g implies continuity of K with respect to both variables at each other point of X X . Moreover, if X is a metric space (or, more generally, a uniform topological space), the continuity is uniform on any subset def. XM Dfx 2 X W K.x; x/ M g:. (1.4). It is obvious that the sum of positive-definite kernels is positive definite too; it does not matter whether the sum is finite or infinite; hence, also the integral of “measurable family” of kernels with this property is positive definite. Rather less obvious is why the product of positive-definite kernels is also positive definite. It follows from the Schur lemma concerning positive definiteness of the Schur product of two positive-definite finite matrices. Furthermore, the power series with nonnegative coefficients of positive-definite kernels is positive definite if the image of the kernel as a function fits in the disk of convergence of the series. In particular, the kernel eK W .x; y/ ! eK.x;y/ ; is positive definite if K is positive definite. The simplest example of the kernel that is automatically positive definite is KW X X 3 .x; y/ ! f .x/f .y/ 2 C;. (1.5). where f is an arbitrary function on X . Despite its simplicity, the kernel (1.5) turns out to be generic in a sense. As shown in Corollary 1, every positive-definite kernel “factorizes” in a properly understood manner through a Hilbert space like that given by formula (1.5); in (1.5), the dimension of the Hilbert space in question is just 1..

(26) 1 The Reproducing Kernel Property and Its Space: The Basics. 7. Uniqueness As already mentioned, each of (A) and (B) is the starting point in getting the other; the other member is determined uniquely. More precisely, given H with two kernels satisfying (1.1), condition (1.3) causes the kernels to coincide. On the other hand, density of linfKx W x 2 X g makes the space H unique when K is given.. Further Properties The list of properties can be extended as follows: 1 (C) If .fn /1 nD0 is a convergent sequence in H, then the sequence .fn .x//nD0 of numbers is convergent for all x 2 X ; the convergence is uniform on all subset XM of the form (1.4).. This is an abstract version of what is known for functions of complex variable: L2 convergence of holomorphic function (via integral representation of this or another kind) forces local uniform convergence. (D) If f is in H, then there is C > 0 such that. j. N X i D1. f .xi /i j2 C 2. N X. K.xi ; xj /i N j ; x1 ; : : : ; xN 2 X; 1 ; : : : N 2 C:. i;j D1. (1.6) Conversely, if there is C > 0 dependent on f and such that (1.6) holds, then f is in H; moreover, kf k is equal to the smallest constant C such that (1.6) holds. Property (D), called RKHS test, is a material hallmark. It allows, for instance, to decide when an arbitrary function on X becomes a member of H. Interestingly enough, it guarantees in particular holomorphicity of a function from the growth condition it satisfies; see (E) A sequence .fn /1 nD0 of functions from H is weakly convergent in H if and only if 1 sequence of norms fn k/1 nD0 is bounded and .fn .x//nD0 converges for all x 2 X . Of course, in such a case, the limit must belong to H. This fact is in [21]. It is also worthy to pay attention to the following simple fact: (F) For x 2 X , K.x; x/ D 0 ” Kx D 0 ” f .x/ D 0 for all f 2 H..

(27) 8. F.H. Szafraniec. Completion of Function Spaces There is no doubt that one of the nicest features of reproducing kernel spaces is the possibility of realization of the completion also as a space of functions on the same set. The next theorem is close to the moment from which we will start thinking about reproducing kernel space as a Hilbert space. Theorem 1. For an inner product space D of complex functions on X , consider the following conditions: (a) There exists (necessarily, exactly one) space H with reproducing kernel K on X such that H is a Hilbert space and D is dense in H; (b) D has reproducing kernel K such that DK is dense in D; (c) Space D has two properties: .c1 / Condition (A) holds on D, .c2 / For each sequence .fn /1 nD0 , which is Cauchy in D, convergence of fn .x/ ! 0 for all x 2 X implies kfn k ! 0. Then (a) and fKx W x 2 X g D H) (b) with the same K, (b) H) (a) with the same K, and (a) ” (c). If (b) or (c) holds, the space H in the condition (a) is unique. Theorem 1 will be often in use. For the time being, let us state the formal definition: unitary space with reproducing kernel will be called reproducing kernel Hilbert space if this unitary space is a Hilbert space. Part (c) of Theorem 1 suggests that not every unitary space for which all functionals x are continuous can be embedded as a dense subset of a reproducing kernel Hilbert space. Indeed it is so, as the example ./ to come shows. However, the following simple fact is worth noting: (d) Subspace H1 of functions on X which is a closed subspace of a Hilbert space H with reproducing kernel on the same X is always a space with reproducing kernel on X , and the reproducing kernel K1 is given by the formula K1 .x; y/ D hPKy ; PKx i;. x; y 2 X;. where P is an orthogonal projection H on H1 and K is a reproducing kernel of the space H. The kernel functions are related by .K1 /x D PKx ;. x 2 X:. Remark 3. A useful observation is in order. Suppose that a unitary space D with the norm k k satisfies (A). If there is another norm k k1 in D, such that kf k kf k1 , f 2 D, then the space .D; k k1 / also satisfies (A)..

(28) 1 The Reproducing Kernel Property and Its Space: The Basics. 9. Hilbert Space of Functions Versus Reproducing Kernel Two fundamental ways of associating couples, Hilbert space and the kernel, will be presented. As a matter of fact, all others reduce to one of these two.. From H to K This way is rather straightforward. Suppose H is given, condition (A) implies existence (Riesz representation theorem) of the family fKx W x 2 X g such that f .x/ D ˚x .f / D hf; Kx i;. x2X. which after defining def. K.x; y/ DhKy ; Kx i;. x; y 2 X. (1.7). provides with the reproducing kernel K. The kernel K can be defined alternatively as K.x; y/ D .˚y / ˚x because the Hilbert space adjoint .˚x / of the functional (=operator) ˚x acts as .˚x / D Kx , 2 C. There is another way, also being used, of constructing a kernel if H is given. It starts from an arbitrary orthonormal basis .e˛ /˛2A in H; here, P card A is just the Hilbert space dimension of H. The Parseval identity yields ˛2A e˛ .y/e˛ is convergent (the meaning of summation in the uncountable case is well explained in [12], remark after Th. 4.16 and Corollary (Bessel inequality)) in H. We arrive at the next property of a reproducing kernel Hilbert space: (G) Reproducing kernel Hilbert space H has exactly one kernel K. In particular for any orthonormal and complete set .e˛ /˛2A in H one has K.x; y/ D. X. x; y 2 X. e˛ .x/e˛ .y/;. ˛2A. and in the topology of the norm of the space H; Kx D. X ˛2A. e˛ .x/e˛ ;. x 2 X:. (1.8).

(29) 10. F.H. Szafraniec. Moreover, f .x/ D. X ˛. hf; e˛ ie˛ .x/;. x 2 X:. (1.9). Equality (1.9) reminds a Fourier series for f . However, in its current, reproducing kernel version, it gives all the values of the function f , instead of the abstract object f itself, as the Fourier expansion does. The formula (1.8) ought to be called Zaremba decomposition or expansion – cf. Comments.. From K to H This construction can be done in a few steps. First is to show that h ; i can be extended in an Hermitian linear way from def. hKy ; Kx i D K.x; y/;. x; y 2 X. to linfKx W x 2 X g, and it becomes an inner product (crucial thing is to prove that hf; f i D 0 H) f D 0) with the reproducing property (B) being automatically satisfied due to definition of K. Then Theorem 1 enters the scene resulting in H to be still a space of functions on X . This feature is important because it allows to avoid getting H as a rather abstract object (two times passing to quotient spaces), which a lot of people do causing a reader to get headache. The gentle approach sketched above will be used, for instance, when constructing tensor products of Hilbert spaces. For further reference, it is convenient to draw the following: Corollary 1. If K is a positive-definite kernel on X , then there exists a Hilbert space H and a map X 3 x 7! Kx 2 H such that H D clolin fKx W x 2 Xg; K.x; y/ D hKy ; Kx i;. x; y 2 X:. The Hilbert space H can be realized as a space of functions on X ; thus, it is also a space with the reproducing kernel K. This corollary is present in the literature as the theorem on the factorization of positive-definite kernels and is connected with many different names (see Comments). Unfortunately, the factorization theorem is usually presented without the last sentence though the charm is just in it. Not only the charm but also the method of proving is subtly related to that sentence..

(30) 1 The Reproducing Kernel Property and Its Space: The Basics. 11. Example 1. It has to be pointed out that a choice among all possible equivalent Hilbert space norms allowed for a particular space H turns out to be significant for appreciating the reproducing property; in particular, it is important when both the reproducing property and uniqueness questions have to be correctly understood. A simple example of couples ..H; h ; i/; K/ and ..H; ch ; i/; c 1 K/ shows what could happen if the reproducing property is not of major concern; they are (trivially) different couples of reproducing kernel Hilbert spaces as the reproducing properties do not coincide. Remark 4. With respect to Theorem 1 in cases where X is a topological space, the following conditions are equivalent: – All functions f in space H are continuous. – Kernel K is continuous as a function of two variables.. Two Constructions of a Positive-Definite Kernel There are essentially two constructions which allow building positive-definite kernels, related to (B) of the previous section.. (B1 ) Kernel from the Would-Be Basis The starting point is the condition (G), a kernel appearing in the formula (1.8). The succeeding question is whether the family of functions in the definition (now it becomes a definition) (1.8) will a posteriori be a basis. Let then .f˛ /˛2A be a family of functions on X such that X ˛2A. jf˛ .x/j2 < C1;. x 2 X:. (1.10). Then the kernel defined by def. K.x; y/ D. X ˛2A. f˛ .x/f˛ .y/;. x; y 2 X;. (1.11). is positive definite, and due to (B), there is the suitable Hilbert space, which makes a couple with K. Proposition 2. For an arbitrary sequence D .˛ /˛2A z `2 .A/, the series X. ˛ f˛ .x/. ˛2A. converges absolutely for every x; the function f W x !. X ˛2A. ˛ f˛ .x/.

(31) 12. F.H. Szafraniec. P is in H with kf k kk`2 .A/ , and also the series ˛2A ˛ f˛ converges in H to f . P In particular, the series ˛2A f˛ .x/f˛ converges in H to Kx , functions f˛ , ˛ 2 A, are in H, and kf˛ k 1. From the following formula comes out of (1.11) k. N X. ! k`2 .A/ D k. f˛ .xi /i. i D0 def. This lets, by putting E D operator V WE 3. N i D0. N X. i Kxi kH :. i D0. ˛2A. nP. N X. f˛ .xi /i. ˛2A. ! !. f˛ .xi /i. i D0. o W x 2 X , to extend an isometric. N X. i Kxi 2 H;. i D0. ˛2A. with preserving the notation, to the operator which, by Corollary 1, is surjection from clo E to H and thus is a unitary operator. The second operator which can be defined is X W W `2 .A/ 3 ! ˛ f˛ 2 H: ˛2A. From the inequality (1.6), it follows that W is a well-defined contraction. Moreover, W. N X. !. D. N X. i. i D0. D. f˛ .xi /i. i D0. !. ˛2A. N X X ˛2A. ! f˛ .xi /i f˛. i D0. N X X .f˛ .xi //f˛ D i K x i ; i D0. ˛2A. and so W D V ;. 2 E:. Further, for 2 `2 .A/ E, we have D. 0 D ; .f˛ .x//˛2A. E `2 .A/. D. X ˛2A. ˛ f˛ .x/ D. * X ˛2A. + D hW ; Kx iH ;. ˛ f˛ ; Kx H. which means that W D 0. Thus, W is a partial isometry in `2 .A/ with the initial space E..

(32) 1 The Reproducing Kernel Property and Its Space: The Basics. 13. Unitarity of W , i.e., equality E D `2 .A/, is equivalent to orthonormality of the sequence .f˛ /˛2A . Proposition 3. The sequence .f˛ /˛2A is always complete. Moreover, the following two conditions are equivalent P (i) 2 `2 .A/ and ˛2A ˛ f˛ .x/ D 0 for each x 2 X implies D 0; (ii) Sequence .f˛ /˛2A is orthonormal in H. Proposition 3 matches the item (G).. (B2 ) Kernel from the Transformation This is a simple matter: suppose for an arbitrary Hilbert space H, a function W X 3 x ! fx 2 H is given. The goal is to obtain a reproducing kernel Hilbert space, which is isomorphic to a subspace of the space H generated by the set ffx W x 2 Xg. More precisely, the construction looks as follows: we define kernel K as def. K.x; y/ Dhfy ; fx iH ;. x; y 2 X;. (1.12). and by H, we denote its Hilbert space. For f 2 H, define ff W X 2 x ! ff .x/ 2 C, the transform of the function f, by formula def. ff .x/ Dhf; fx i;. x 2 X:. (1.13). It turns out that the image H of the transformation TW f ! ff of the space H corresponding to the map by the formula (1.13) is identical with H. From the RKHS test (D) it follows immediately that the functions in H are in H and additionally that T is a contraction. That kernel functions Kx are in H , it follows straightforwardly from (1.12) and (1.13). It is so indeed, from (1.7), (1.12) hKy ; Kx iH D hfy ; fx iH ;. x; y 2 X. (1.14). which, together with (1.13), is enough for Kx D ffx ;. x 2 X:. (1.15). This together with the formula (1.14) implies that T is an isometry on the linear span of vectors fx , x 2 X , and naturally on its closure. Density of kernel functions gives.

(33) 14. F.H. Szafraniec. the same formula as (1.14) for arbitrary functions from H. Therefore, an identification of the space H with reproducing kernel K with the space H is established. Formula (1.14) states that transformation T is a unitary operator which maps clolinffx W x 2 X g to H but as an operator on H it is a partial isometry. If P denotes the orthogonal projection from H to clolinffx W x 2 X g, ff D fP f ;. f 2 H:. For the sake of future reference, itemize the following fact. Corollary 2. H is unitarily isomorphic, by the above construction, with H if and only if clolinffx W x 2 X g D H: Example 2. A simple, almost trivial example of application of the method (B2 ). Let H be an arbitrary Hilbert space. Take X D H and D idX . In order to examine carefully the method described above, the trivial identification H D H has to be accepted, which in turn forces a certain ambiguity in referring to elements from H D H. Namely, f 2 H and f 2 H are the same element that, depending on the context, is written down in a different manner, and the formula ff D f documents this identification – this is the only difficulty (of rather psychological nature) to be overcome in this example. It becomes clear how to understand the formula (1.13). It, taking into account (1.15), has the form Kf .g/ D hff ; fg iH D hf; giH ;. g 2 H:. As a conclusion, one gets that, after identification f. ! Kf ;. f 2 H;. every Hilbert space is a RKHS with the reproducing kernel defined on X D H. The kernel K is just the function .f; g/ 7! hf; giH . This looks like a trivial observation, but it is far from being so; its consequences are serious and remarkable..

(34) 1 The Reproducing Kernel Property and Its Space: The Basics. 15. Operations on Reproducing Kernel Spaces Comparisons of Reproducing Kernels Given two kernels K1 and K2 , both positive definite on X , write K1 K2 if K2 K1 is positive definite on X . Let H1 and H2 be respective Hilbert spaces. (e) Following conditions are equivalent: (e1 ) K1 K2 . (e2 ) H1 H2 and kf kH2 kf kH1 for f 2 H1 . (e3 ) There exists a contraction T1 2 B.H2 ; H1 /, such that K1 .x; y/ D hT1 .K2 /y ; T1 .K2 /x iH1 ;. x; y 2 X:. (1.16). (e4 ) There exists a contraction T2 2 B.H2 /, such that K1 .x; y/ D hT2 .K2 /y ; T2 .K2 /x iH2 ;. x; y 2 X:. (1.17). In such cases: (e5 ) There exists a contraction T0 , such that T0. X. def Ni .K2 /xi D. i. X. Ni .K1 /xi ;. xi 2 X;. i. which satisfies .e3 /, and if V is an inclusion operator H1 in H2 , the following holds . . . V T0 D T0 T0 D T1 T1 D T2 T2 : Remark 5. Linear subspace H1 , mentioned in (e), does not have to be closed in H2 (i.e., in the norm H2 ); inequality of norms in (D) tells us that inclusion V of the space H1 in H2 is a mere contraction; if it is an isometry, we are in the situation (d). However, we are obliged to note (the closed graph theorem may be used) the following fact: (e6 ) A linear subspace H1 is closed in H2 if and only if the condition .e2 / is completed with the inequality kf kH1 c kf kH2 for f 2 H1 with some positive constant c. Example 3. Let H be `2 . For a sequence a D .an /1 nD0 given by a2n D n and def a2nC1 D 1, n D 0; 1; : : : def.

(35) 16. F.H. Szafraniec. def. K1 .m; n/ D. 1 ım;n ; an. m; n D 0; 1; : : :. is positive definite. Then, the space H1 given by K1 is not a closed subspace of `2 . Remark 6. Going back to the issue of uniqueness, mentioned in Remark 1, the following fact has to be pointed out. The following conditions are equivalent: – Couples ..H; h ; i1 /; K1 / and ..H; h ; i2 /; K2 / are reproducing ones. – There exist constants c and d such that ck k1 k k2 d k k1 . – There exist constants c 0 and d 0 such that c 0 K1 K2 d 0 K1 . A similar situation to the one presented in (e) is included in the following fact: Proposition 4. Let H and K be Hilbert spaces and T W H ! K be a contraction. If H is a Hilbert space with reproducing kernel K on X , then for the kernel K1 defined by def. K1 .x; y/ DhTKy ; TKx iK ;. x; y 2 X. one has K1 K. Moreover for .H1 ; K1 / and .H2 D H; K2 D K/, observations made in (e) hold.. Sum and Subtraction of Reproducing Kernels Previous considerations have laid fundaments for the material covered in this subsection. If K D K1 C K2 , then, since Ki K, i D 1; 2, each of these kernels fits the situation described by (e1 ) (all that is needed is a slight modification of the notation). Going further, there are Hi which are linear subspaces H corresponding to K. Moreover, the following decomposition holds: kf k2H D kT11 f k2H1 C kT21 f k2H2 ; where Ti 1 is an operator which makes an appearance in (e3 ) and corresponds to the pair of kernels Ki K, i D 1; 2. If the function Hilbert space H decomposes to H D H1 ˚H2 , where Hi , i D 1; 2 are closed subspaces H, which, seen as spaces on their own with the norm induced by H, are Hilbert spaces with the reproducing kernel Ki , then all that is needed to start from (e3 ) is to observe that the second condition is automatically satisfied..

(36) 1 The Reproducing Kernel Property and Its Space: The Basics. 17. Product of Reproducing Kernels: Tensor Product of Hilbert Spaces This is where arrangement and its attractive consequences are spectacular for those for whom the esthetic side is as important as reaching a mathematical goal itself. For simplicity, it is enough to consider the case of two: the sets X1 and X2 , positive-definite kernels K1 and K2 , and the spaces H1 and H2 . The kernel on X1 X2 introduced by def. K.x1 ; x2 ; y1 ; y2 / D K1 .x1 ; y1 /K2 .x2 ; y2 /;. x1 ; y1 2 X1 ; x2 ; y2 2 X2. is called the product kernel (this is related to the Kronecker product of matrices) of K1 and K2 . The kernel K is positive definite which stems from Schur’s lemma already mentioned here. Let H denote the resulting Hilbert space of functions on X1 X2 . The kernel functions look like Kx1 ;x2 D K. ; ; x1 ; x2 / D .K1 /x1 .K2 /x2 ;. x1 2 X1 ; x2 2 X2. giving rise to def. .K1 /x1 ˝ .K2 /x2 D Kx1 ;x2 .x1 ; x2 /;. xi 2 Xi ; i D 1; 2:. Because the mapping .f1 ; f2 / 7! f1 ˝ f2 is bilinear, with D1 and D2 being the linear span of respective kernel functions, linff1 ˝ f2 W f1 2 D1 ; f2 2 D2 g is precisely nothing but a concrete and simple accomplishment of the algebraic tensor product D1 ˝ D2 of D1 and D2 designed as a space of functions on X1 X2 , de facto a subspace of H.. What is H? The answer is simple: because for f ? D1 ˝ D2 0 D hf; .K1 /x1 ˝ .K2 /x2 i D f .x1 ; x2 /;. .x1 ; x2 / 2 X1 X2 ;. f D 0 and, therefore, D1 ˝ D2 is dense in H. Thus, H, as a completion of D1 ˝ D2 , becomes the Hilbert space tensor product of H1 and H2 , and, formally, the couple .K; H/ is such for the couples jr .K1 ; H1 / and .K2 ; H2 /. Because for fi 2 Di and xi 2 Xi hf1 ˝ f2 ; .K1 /x1 ˝ .K2 /x2 i D f1 .x1 /f2 .x2 / D hf1 ; .K1 /x1 iH1 hf2 ; .K2 /x2 iH2 ; looking at particular equalities in there, the following comes out: 1o The mapping H1 D2 3 .f1 ; f2 / 7! f1 ˝ f2 2 H is continuous. 2o For f1 ; g1 2 H1 i f2 ; g2 2 H2 , there is hf1 ˝ f2 ; g1 ˝ g2 iH D hf1 ; g1 iH1 hf2 ; g2 iH2 :.

(37) 18. F.H. Szafraniec. Let Hi , i D 1; 2, be arbitrary Hilbert spaces. As already pointed out in Example 2, each of them can be identified with a reproducing kernel Hilbert space def with the kernel Ki .xi ; yi / Dhxi ; yi iHi , xi ; yi 2 Hi . In this way, one can apply the above construction getting a concrete realization of the Hilbert space tensor product H1 ˝ H2 as a space of functions on H1 H2 ; the same refers to the algebraic tensor product D1 ˝ D2 . This is a beautiful example of how useful the theory can be. The above construction can be extended to an arbitrary finite number of spaces, n which gives in particular H˝ , where H is a Hilbert space, and leads in an intriguing way to constructions of different kinds of Fock spaces as done in [10].. Restrictions of Kernels and Interpolation in a Reproducing Kernel Hilbert Space So far, all kernels on which operations were acting were defined on the same set X . From now on, this will no longer be the case. For a subset X1 of the set X , we will use the following notation: def. H0 Dff 2 HW f .x/ D 0; x 2 X1 g;. def. H1 D clolinfKx W x 2 X1 gI. P0 and P1 are orthogonal projections on H0 and H1 , respectively. From the reproducing property (1.1) and condition (d), it follows that H0 is a closed subspace of H; moreover, H D H0 ˚ H1 :. (1.18). This is an opportunity to define two notions, which correspond to two extreme cases for which the decomposition (1.18) can be considered. Set Y X will be called set of uniqueness of the set A CX , if f; g 2 A and f .x/ D g.x/ for x 2 Y implies f D g. Family of all sets of uniqueness will be def denoted by UA , and set of zeroes of the space H by ZK Dfx 2 X W K.x; x/ D 0g; these names are self-explanatory. Thus, H0 D f0g ” X1 2 UK ;. H1 D f0g ” X1 ZK ;. (1.19). making use of (F). Moreover, If K1 K2 , then ZK2 ZK1 and UH2 UH1 : Suppose that besides a positive-definite kernel K on X also set Y and a map W Y ! X are given. This allows to define new kernel K , this time on Y , by def. K .y1 ; y2 / D K..y1 /; .y2 //;. y1 ; y2 2 Y:. (1.20).

(38) 1 The Reproducing Kernel Property and Its Space: The Basics. 19. This kernel is positive definite; suitable Hilbert space will be denoted by H . From the RKHS test, infer that a linear operator T W H 3 f ! f ı 2 H ; by the definition taking DK onto (it is so because .K /y D K.y/ ı for each y 2 Y ) DK extends to the whole space H kf ı kH inffkgkH W g ı D f ı g;. (1.21). which means that T is a contraction. It turns out that in the formula (1.21), the equality holds also for functions f from the linear span of kernel functions and thus for any function f in H. As a result, we have that T is a surjective contraction. Let now Y be denoted by X1 and be a subset X . If is an inclusion, in the above notation, we replace index (at the bottom) by X1 . Obtained in this manner, that is, by the formula (1.20), kernel KX1 can be called the restriction of the kernel K to the subset X1 and similarly for the space HX1 (notice that elements HX1 are functions on X1 , not – as elements H1 – functions on the whole X ; the same holds also with respect to the kernel KX1 ). It is easy to see that T1 is injective if and only if X1 2 UH , which simplifies the formula (1.21) to isometric equality, making T1 a bijection. The first of conditions (1.19) tells us also that restriction of the kernel to the set of uniqueness does not modify the Hilbert space, that is, the first element of the pair. For the more complete picture, take a slightly different look at the previous considerations. Arbitrary function f from H decomposes, in accordance with (1.18), as f D f0 C f1 , where for the restrictions simple formula f jX1 D f1 jX1 ;. f 2H. happens. This means that the kernel KX1 is related to the kernel K with the orthogonal projection P1 , that is, for all x; y 2 X1 , KX1 .x; y/D hP1 Ky ; P1 Kx iH and .KX1 /x D P1 Kx jX1 . The interpolation result which follows is a kind of a generalization of the RHKS test. It concerns interpolation within H, and do not get confused with interpolation á la Pick-Nevanlinna, which by the way, is related to Szaf . . . . Theorem 2 (interpolation). Let X1 be a subset of the set X and .H; K/ a couple with reproducing kernel on X . For a given function f1 W X1 ! C, there exists a function f 2 H such that f1 D f jX1 and only if for some C (dependent on f1 ) j. N X i D1. f1 .xi /i j2 C 2. N X i;j D1. K.xi ; xj /i N j ;. x1 ; : : : ; xN 2 X1 ; 1 ; : : : N 2 C:.

(39) 20. F.H. Szafraniec. If this happens, f1 must belong to H1 (restriction of the space H to X1 ), and the norm kf k is not greater than C .. An Instructive Example ./ Have a look at the things yet to come and consider a Bergman space B 2 .D/, that is, subspace of all holomorphic functions in D belonging to L2 .D; 1 m2 /, where m2 is the two-dimensional Lebesgue measure. Bergman space has the reproducing kernel of the form K.z; w/ D. 1 : .1 zw/ N 2. Take the sequence .zi /1 i D1 of points in D such that 1 X .1 jzi j/ < C1;. zi 6D 0:. i D1. Denote the range of this sequence by X1 , and for the sake of symmetry, think of D as X . It is known that there exists nonzero functions in B 2 .D/ equal to zero on X1 ; an infinite Blaschke product may provide such example: def. B.z/ D. 1 Y zNi z zi : jz j 1 zNzi i D0 i. This means that X1 62 UB2 .D/ , that is, the decomposition (1.18), is nontrivial. On the other hand, however, X1 2 UP.X / , which gives the possibility of carrying over the norm from B 2 .D/ to P.X1 / while preserving the condition (c1 ) in Theorem 1 for such a space P.X1 / normed in this manner. However, the condition (c2 ) is not satisfied. It is enough to take a sequence of polynomials .pn /1 nD1 convergent in B 2 .D/ to Blaschke product B. Such a sequence exists, since P.X / is dense in B 2 .D/ (recall that for polynomials, the norms B 2 .D/ and P.X1 / are equal). 2 Then .pn /1 nD1 is a Cauchy sequence in P.X1 /; from (C) in B .D/, it follows that pn .zi / ! B.zi / D 0 for all i , but kpn k ! kBk > 0. In the above, the notation Pd .X / is used for the set of all polynomials understood as functions on X Cd , def P.X / D P1 .X /. The moral is that the above “double restriction operation” (i.e., first to a dense subspace then to subset) performed in a reproducing kernel Hilbert space may not lead back to the reproducing kernel space (the reason may be that the kernel function is not included in D). From this example, the following observation can be deduced, which complements what has been said so far..

(40) 1 The Reproducing Kernel Property and Its Space: The Basics. 21. Proposition 5. Let .H; K/ be a reproducing couple on X . For a linear subspace def D H such that X1 X is its set of uniqueness, D1 Dff jX1 W f 2 Dg is an inner product space with reproducing kernel only if Kx 2 D for x 2 X1 . The example above should serve as a warning: always have in mind the second element of the couple which, in accordance with our convention, is a reproducing kernel Hilbert space. Limit of Spaces with Reproducing Kernel Let the sequence .Xn /1 nD1 of subsets of some set (there is no need to pinpoint it) satisfying def. XD. 1 \. Xn 6D ¿:. nD1. Let also be given a sequence of couple .Hn ; Kn / on Xn , n D 0; 1; : : : Suppose that def. K.x; y/ D lim Kn .x; y/ < C1; n!1. x; y 2 X:. The kernel K is positive definite, and together with its Hilbert space, H is a limit of pairs ..Hn ; Kn //1 nD1 . The basic question is: what is the relation between the space Hn and the space H? If f 2 Hn , then, due to (D), ˇ2 ˇN N ˇ ˇX X ˇ ˇ f .x / kf k Kn .xi ; xj /i N j ; ˇ i iˇ Hn ˇ ˇ i D1. xi 2 Xn ; i 2 C; i D 1; 2; : : :. i;j D1. Taking as the limit in the evaluation, the RKHS test (D) leads directly to the statement: if there exists a subsequence .Xnk /1 kD1 such that f 2 Hnk and lim inf kf kHnk < C1, then f jX belongs to H. When all Xn are equal to X and the sequence .Kn /1 nD1 is increasing, that is, Kn KnC1 ;. n D 1; 2; : : : ;. condition (e2 ) gives us an inclusion Hn HnC1 and boundedness of the sequence of norms kf kHn kf1 kH1 . Thus, Hn HnC1 H;. n D 1; 2; : : :.

(41) 22. F.H. Szafraniec. Finite-Dimensional Spaces Appearance of this topic here might be slightly surprising. As we will see soon enough, this is being done on purpose. Let H be a finite-dimensional space, say dim H D d . Take sequence of linearly independent functions .fn /dnD1 from H. The scalar product in H has the form hf; gi D. d X. ˛i;j i N j ;. f D. i;j D1. d X. i fi ; g D. i D1. d X. j gj ;. j D1. where matrix .˛i;j /di;j D1 is positive definite. In fact, it is Gram matrix of the sequence .fn /dnD1 , that is, ˛i;j D hfi ; fj i;. i; j D 1; : : : ; d;. (1.22). and as such, it is non-singular. Denote by .ˇi;j /di;j D1 the matrix adjoint to .˛i;j /di;j D1 , that is, such that d X. ˛i;k ˇNj;k D ıi;j ;. i; j D 1; : : : ; d:. (1.23). kD1. The matrix .ˇi;j /di;j D1 is also positive definite, and def. K.x; y/ D. d X. ˇi;j fi .x/fj .y/;. x; y 2 X. (1.24). i;j D1. is a positive-definite kernel. Thus, a Hilbert space H with the scalar product given by (1.22) is the space with reproducing kernel given by (1.24), and (vice versa) every finite-dimensional space in which .fn /dnD1 is a linearly independent sequence and the kernel K is given by (1.24) with a positive-definite matrix .ˇi;j /di;j D1 is a reproducing kernel Hilbert space whose scalar product is given by (1.22) through (1.23). A particular example is the case when the matrix .˛i;j /di;j D1 , and consequently .ˇi;j /di;j D1 , is the identity matrix. Then the situation described in (G) happens. Look now at another example. Of course, all finite-dimensional Hilbert spaces are isomorphic with Cd , which makes the question whether our efforts so far are futile. Take a sequence of polynomials .pn /1 nD0 of one variable with real coefficients, ordered by the condition deg pn D n:. (1.25).

(42) 1 The Reproducing Kernel Property and Its Space: The Basics. 23. This sequence is called an orthogonal polynomial one, if there exists a Borel measure on R such that Z R. pm pn d D Nn ım;n ;. m; n D 0; 1; : : : ;. where all numbers Nn are positive. If all Nn equal 1, polynomials are orthonormal, 2 which means that the sequence .pn /1 nD0 is orthonormal in L .R; /. It does not have to be complete, and moreover, the set ( z 2 CW. 1 X. ) jpn .z/j < C1 2. (1.26). nD0. might be empty. It is not empty if the sequence is finite (like for Krawtchouk polynomials, for instance). If .pn /1 nD0 is a sequence of orthogonal polynomials, then the Hilbert space of polynomials of degree at most N , in which .pn /N nD0 is an orthonormal basis, will be denoted by PN . Ignoring the set (1.26), the kernel def. KN .x; y/ D. N X. pn .x/pn .y/;. x; y 2 R or x; y 2 C. nD0. is always well defined. The couple .PN ; KN / is an N -dimensional reproducing kernel Hilbert space. From the general theory of orthogonal polynomials, it is known that the sequence .pn /1 nD0 which satisfies (1.25) is a sequence of polynomials orthogonal with respect def to a measure supported by infinitely many points if and only if it satisfies p1 D 0 and the so-called three-term recurrence relation Xpn D an pnC1 C bn pn C an1 pn1 ;. def. a1 D 1;. n D 0; 1; : : :. with an 6D 0 and bn 2 R for all n. From this relation, an alternative formula for the kernels KN follows ( KN .x; y/ D. aN .pN C1 .x/pN .y/ pN .x/pN C1 .y//.x y/1. when x 6D y. aN .pN0 C1 .x/pN .x/. when x D y. . pN0 .x/pN C1 .x//. ;. which is connected with the names of Christoffel and Darboux. This formula significantly simplifies using the reproducing property p.x/ D hp; .KN /x i;. deg p N; x 2 C;. since all it requires is to know two Fourier coefficients hp; pN C1 i i hp; pN i..

(43) 24. F.H. Szafraniec. The natural question whether the reproducing property can be extended to all polynomials is, as we know, directly related to the question whether the set (1.26) is nonempty. It turns out that the question is settled by related orthogonal moment problem. Now some definitions. Sequence .an /1 nD0 is called sequence of Hamburger moment if there exists a nonnegative measure on R such that Z an D. t n .d t/;. n D 0; 1; : : : I. R. the moment problem is determinate if a measure representing the sequence .an /1 nD0 on R is unique; otherwise, the problem is called indeterminate. The basic fact concerning the Hamburger moment is the following: Theorem 3. Sufficient conditions for the moment problem to be indeterminate is that at least one z 2 C belongs to the set (1.26), and the necessary condition is that all z 2 C belong to this set. For an indeterminate moment problem, the kernel def. K.x; y/ D. 1 X. pn .z/pn .w/;. z; w 2 C. nD0. is well defined and its Hilbert space is a space of entire functions. An interesting observation is that this space may not fill up the whole of L2 ./, where is a measure representing orthogonality. Remark 7. Note, as a sort of digression, that the space corresponding to the kernel (1.5) is one dimensional, that is, equal to Cf .. Multiplication Operators and Multipliers Multiplication operators and multipliers are two different faces of the same problem. Let begin with the definition. Fixing a Hilbert space H with reproducing kernel K, take 'W X ! C and put def. D' Dff 2 HW 'f 2 Hg: Operator M' W D' ! H will be called multiplication operator by ' when D' is dense in H. For a multiplier, distinguish two (not necessarily disjoint) situations: for D' D H, the function ' is called multiplier of a couple .H; K/; and if D' is merely dense in H, ' is said densely defined multiplier..

(44) 1 The Reproducing Kernel Property and Its Space: The Basics. 25. Proposition 6. M' is always a closed operator. As a consequence, if ' is a multiplier, M' is a bounded operator. Conversely, if M' is a bounded operator, then j'.x/j kM' k provided K.x; x/ > 0. An important issue in these investigations is to determine the operator adjoint of M' . Consider first the simple fact: Proposition 7. For every x 2 X hM' f; Kx i D hf; '.x/Kx i: . . If M' is densely defined, then Kx 2 D.M' / and M' Kx D '.x/Kx for every x 2 X. More information about these matters is included in the following theorem: Theorem 4. For an operator A which is densely defined in H and a complex function ' on X following conditions as equivalent: . . (a) DK D.A / and A Kx D '.x/Kx for x 2 X ; (b) Operator A is closable and AN M' . . If such an operator A exists, then M' is densely defined and M' Kx D '.x/Kx for every x 2 X . If M' is densely defined, then (a) holds with A D M' . In many situations which happen the following fact might be satisfactory. Corollary 3. M' is a bounded operator on H, that is, M' 2 B.H/ if and only if there exists an operator R bounded on Dk , that is, R 2 B.DK /, such that RKx D '.x/Kx for every x 2 X . Then R D M' and kM' k D kRk. An observation related to Theorem 4 might also be useful. Proposition 8. Suppose that a constant function 1X W X 3 x ! 1 2 C belongs to H. If A is a closable operator such that N \ D.MKx /, DK D.A / \ D.M / and (c) For each x 2 X function 1X 2 D.A/ Kx . N X i; hA Kx ; MKy 1X i D hMKy Kx ; A1. x; y 2 X;. N X. then A satisfies condition (a) of Theorem 4 with ' D A1 Again, there is a milder variant of this fact.. . (1.27).

(45) 26. F.H. Szafraniec. Corollary 4. Suppose that a constant function 1X W X 3 x ! 1 2 C belongs to H and for x 2 X hold 1X 2 D.MKx / and DK D.MKx /. If A 2 B.H/ satisfies (1.27), then A D MA1X . def. With M.D/ Df'W D D' g, the theorem below will be useful in characterization of multipliers, and its proof follows straightforwardly from the RKHS test (D). Proposition 9. Let S be such a subset of H that lin S D D. Function 'W X ! C belongs to M.D/ if and only if 8f 2 S H) 9 C > 0; j. n X. '.xi /f .xi /i j2 C. i D1. n X. K.xi ; xj /i Nj. i;j D1. for any xi in X and i in C. In particular: Corollary 5. If ' 2 M.DK /, then for every y 2 X there exists C such that j'.x/j C jK.x; y/j1 K.x; x/1=2 ;. x 2 X:. Comments Kernels as functions of two variables are usually associated with integral equations; this is also the origin of the definition of positive definiteness; it was given by Mercer in 1909; cf. [3]. A similar concept, in a different context, appeared in the works of [7] in the third decade of the previous century. It was the nonnegative functions defined on the real line, i.e., functions satisfying N X. f .xi xj /i N j 0;. x1 ; : : : ; xN 2 R; 1 ; : : : N 2 C:. i;j D1. Separation of variables xi and xj through the definition def. K.x; y/ D f .x y/;. x; y 2 R. leads to positive-definite kernels, and in this manner, many illustrious constructions can be unified, in particular, the famous GNS construction for C algebras; about all this more will be in the next chapter [22]. Positive definiteness of the kernel K means, in practice, positive definiteness of each of the matrices .K.xi ; xj //N i;j D0 , which also holds for quadratic forms related.

(46) 1 The Reproducing Kernel Property and Its Space: The Basics. 27. to such matrices; in this matter, the terminology somewhat differs – quite often the general case is distinguished from the one which requires additionally that the form equals zero only for the null vector. From the construction presented in part (B), we escape such worries. The RKHS test (D), despite its simplicity, cannot be found stated explicitly in the literature; just in case, we refer to [18] or [19]. In 1907, the paper [23] appeared. The author, professor at the Jagiellonian University in Kraków, considered there boundary value problems for harmonic and biharmonic functions. Among other things, one can find two formulae, that is, (1.1) and (1.8), which occupy the predominant part of this and the following chapter [22]. These two formulae are on p. 170 of [23], a scrap of which has been arranged into the cover page of the book [21] – attached here at the end (notice the formulae are there in the reverse order). Unhappily, Zaremba did not prettify his discovery by giving it special names – fortunately, mathematical discovery has its value regardless of how it is called. The reproducing property of Stanisław Zaremba [23] was “discovered” and strongly acknowledged for the first time in [4]. It should be emphasized that besides the reproducing property (1.1), formula (1.8) appears in [23] for the first time, a good reason to call it after Zaremba. This formula is frequently used along when dealing with the reproducing kernel property. Unfortunately, a large number of authors proclaim priority to discovery of the reproducing kernel to Bergman quoting [5]; this is far from being true. Aronszajn [4] disposed of this myth in a very decisive way. Theorem 1 connects two results of slightly different nature: one is completion of a unitary space which already has a reproducing kernel; the second is completion of a unitary function space to a function space which – this is an additional conclusion made using the same assumptions – is a reproducing kernel space; the connection rests in the fact that the proof is almost the same. The details of the proof of the first fact are usually cast aside and replaced with general remarks. Proof of the second fact, though, is useful when we want to obtain a reproducing kernel Hilbert space from a given unitary space – in such a case, checking that the condition (c2 ) holds is enough. The current presentation of this fact is modeled after [1–3]; by the way, these papers of Aronszajn contain a well-done attempt at building up the theory of RKHSs. Concerning Corollary 1 the literature mentions mostly Kolmogorov; however, the proper reference has to be to [9]. The procedure exhibited in (B2 ) is attributed in [6] to Loève. Saitoh picked up this possibility of defining RKHSs in [13] and develop it further in [14] as well as in a number of papers. Theorem 3 is a classical result; look at [15]; for modern presentation, see [16]. Multipliers are well known in the context of the operator theory; see [8]. Densely defined multipliers can be found in [11]; see also [19, 20]. Needless to say that the large part of Chapter 1 of [21] and subsequently of the present chapter is inspired by [3]..

(47) 28. F.H. Szafraniec. Notes. This is a front cover of the monograph by F.H. Szafraniec, Przestrzenie Hilberta z ja¸drem reprodukuja¸ cym (Reproducing kernel Hilbert spaces, in Polish), Wydawnictwo Uniwersytetu Jagiellońskiego, Kraków, 2004, which exposes the crucial formulae of Zaremba. Some information about Stanisław Zaremba can be found here: http://www-history.mcs.st-andrews.ac.uk/Biographies/Zaremba.html http://info-poland.buffalo.edu/web/sci_health/math/Zaremba/Zaremba.