Conservative Realizations of Herglotz-Nevanlinna Functions
This book is devoted to conservative realizations of various classes of Stieltjes, inverse Stieltjes, and general Herglotz-Nevanlinna functions as impedance functions of linear systems. The main feature of the monograph is a new approach to the realizatio
- PDF / 5,360,229 Bytes
- 540 Pages / 439.37 x 666.142 pts Page_size
- 38 Downloads / 164 Views
Editors: Joseph A. Ball (Blacksburg, VA, USA) Harry Dym (Rehovot, Israel) Marinus A. Kaashoek (Amsterdam, The Netherlands) Heinz Langer (Vienna, Austria) Christiane Tretter (Bern, Switzerland) Associate Editors: Vadim Adamyan (Odessa, Ukraine) Albrecht Böttcher (Chemnitz, Germany) B. Malcolm Brown (Cardiff, UK) Raul Curto (Iowa, IA, USA) Fritz Gesztesy (Columbia, MO, USA) Pavel Kurasov (Lund, Sweden) Leonid E. Lerer (Haifa, Israel) Vern Paulsen (Houston, TX, USA) Mihai Putinar (Santa Barbara, CA, USA) Leiba Rodman (Williamsburg, VA, USA) Ilya M. Spitkovsky (Williamsburg, VA, USA)
Honorary and Advisory Editorial Board: Lewis A. Coburn (Buffalo, NY, USA) Ciprian Foias (College Station, TX, USA) J.William Helton (San Diego, CA, USA) Thomas Kailath (Stanford, CA, USA) Peter Lancaster (Calgary, Canada) Peter D. Lax (New York, NY, USA) Donald Sarason (Berkeley, CA, USA) Bernd Silbermann (Chemnitz, Germany) Harold Widom (Santa Cruz, CA, USA)
Yuri Arlinskii Sergey Belyi Eduard Tsekanovskii
Conservative Realizations of Herglotz–Nevanlinna Functions
Yuri Arlinskii Department of Mathematics East Ukrainian National University Kvartal Molodizhny, 20-A 91034 Lugansk, Ukraine [email protected]
Sergey Belyi Department of Mathematics Troy University Troy, AL 36082, USA [email protected]
Eduard Tsekanovskii Department of Mathematics Niagara University, NY 14109, USA [email protected]
2010 Mathematics Subject Classification: 47A, 47B ISBN 978-3-7643-9995-5 e-ISBN 978-3-7643-9996-2 DOI 10.1007/978-3-7643-9996-2 Library of Congress Control Number: 2011930752 © Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com
To the memory of Moshe Liv˘sic, a Teacher and a great mathematician
Preface Consider the system of equations ⎧ dχ i + T χ(t) = KJψ− (t), ⎪ ⎪ ⎨ dt χ(0) = x ∈ H, ⎪ ⎪ ⎩ ψ+ = ψ− − 2iK ∗ χ(t),
(0.1)
where T is a bounded linear operator from a Hilbert space H into itself, K is a bounded linear operator from a Hilbert space E (dim E < ∞) into H, J = J ∗ = J −1 maps E into itself, and Im T = KJK ∗ . If for a given continuous in E function ψ− (t) ∈ L2[0,τ0 ] (E) we have that χ(t) ∈ H and ψ+ (t) ∈ L2[0,τ0 ] (E) satisfy the system (0.1), then the following metric conservation law holds: τ τ 2 2 2χ(τ ) − 2χ(0) = (Jψ− , ψ− )E dt − (Jψ+ , ψ+ )E dt, τ ∈ [0, τ0 ]. (0.2) 0
0
Given an input vector ψ− = ϕ− eizt ∈ E, we seek solutions to the system (0.1) as an output vector ψ+ = ϕ+ eizt ∈ E and a state-space vector χ(t) = xeizt in H, (z ∈ C). Substituting the expressions for ψ± (t) and χ(t) in (0.1) allows us to cancel exponential terms and conver
Data Loading...
