Operator Theory: Advances and ApplicationsVol. 171

Editor:I. Gohberg

Editorial Office:School of Mathematical SciencesTel Aviv UniversityRamat Aviv, Israel

Editorial Board:D. Alpay (Beer-Sheva)J. Arazy (Haifa)A. Atzmon (Tel Aviv)J. A. Ball (Blacksburg)A. Ben-Artzi (Tel Aviv)H. Bercovici (Bloomington)A. Böttcher (Chemnitz)K. Clancey (Athens, USA)L. A. Coburn (Buffalo)R. E. Curto (Iowa City)K. R. Davidson (Waterloo, Ontario)R. G. Douglas (College Station)A. Dijksma (Groningen)H. Dym (Rehovot)P. A. Fuhrmann (Beer Sheva)B. Gramsch (Mainz)J. A. Helton (La Jolla)M. A. Kaashoek (Amsterdam)H. G. Kaper (Argonne)

Subseries: Advances in Partial Differential Equations

Subseries editors:Bert-Wolfgang Schulze Universität Potsdam Germany

Sergio AlbeverioUniversität BonnGermany

S. T. Kuroda (Tokyo)P. Lancaster (Calgary)L. E. Lerer (Haifa)B. Mityagin (Columbus)V. Olshevsky (Storrs)M. Putinar (Santa Barbara)L. Rodman (Williamsburg)J. Rovnyak (Charlottesville)D. E. Sarason (Berkeley)I. M. Spitkovsky (Williamsburg)S. Treil (Providence)H. Upmeier (Marburg)S. M. Verduyn Lunel (Leiden)D. Voiculescu (Berkeley)D. Xia (Nashville)D. Yafaev (Rennes)

Honorary and AdvisoryEditorial Board:C. Foias (Bloomington)P. R. Halmos (Santa Clara)T. Kailath (Stanford)H. Langer (Vienna)P. D. Lax (New York)M. S. Livsic (Beer Sheva)H. Widom (Santa Cruz)

Michael DemuthTechnische Universität ClausthalGermany

Jerome A. Goldstein The University of Memphis, TNUSA

Nobuyuki ToseKeio University, YokohamaJapan

Advances inPartialDifferentialEquations

The Extended Field of Operator Theory

Michael A. DritschelEditor

BirkhäuserBasel . Boston . Berlin

Michael A. DritschelSchool of Mathematics and StatisticsUniversity of NewcastleNewcastle upon Tyne NE1 7RUUKe-mail: m.a.dritschel@ncl.ac.uk

Editor:

bibliographic data is available in the Internet at <http://dnb.ddb.de>.

obtained.

To my friend and colleague

Nicholas Young

on his retirement

Contents

Editorial Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

Group Photo for IWOTA 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Talk Titles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

T. AktosunInverse Scattering to Determine the Shape of a Vocal Tract . . . . . . . . . . 1

J.R. ArcherPositivity and the Existence of Unitary Dilationsof Commuting Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

D.Z. Arov and O.J. StaffansThe Infinite-dimensional Continuous TimeKalman–Yakubovich–Popov Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A. Bottcher and H. WidomFrom Toeplitz Eigenvalues through Green’s Kernelsto Higher-Order Wirtinger-Sobolev Inequalities . . . . . . . . . . . . . . . . . . . . . . 73

I. Chalendar, A. Flattot and J.R. PartingtonThe Method of Minimal Vectors Applied toWeighted Composition Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

I. Gohberg, M.A. Kaashoek and L. LererThe Continuous Analogue of the Resultant andRelated Convolution Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

G. Heinig and K. RostSplit Algorithms for Centrosymmetric Toeplitz-plus-HankelMatrices with Arbitrary Rank Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

M. Kaltenback and H. WoracekSchmidt-Representation of Difference Quotient Operators . . . . . . . . . . . . 147

A.Yu. KarlovichAlgebras of Singular Integral Operators with PiecewiseContinuous Coefficients on Weighted Nakano Spaces . . . . . . . . . . . . . . . . . 171

viii Contents

Yu.I. KarlovichPseudodifferential Operators with CompoundSlowly Oscillating Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

E. Kissin, V.S. Shulman and L.B. TurowskaExtension of Operator Lipschitz and CommutatorBounded Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

V.G. Kravchenko and R.C. MarreirosOn the Kernel of Some One-dimensional SingularIntegral Operators with Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

V.S. Rabinovich and S. RochThe Fredholm Property of Pseudodifferential Operatorswith Non-smooth Symbols on Modulation Spaces . . . . . . . . . . . . . . . . . . . . 259

J. Rovnyak and L.A. SakhnovichOn Indefinite Cases of Operator Identities Which Arisein Interpolation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

N. SamkoSingular Integral Operators in Weighted Spaces ofContinuous Functions with Oscillating Continuity Moduliand Oscillating Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

N. VasilevskiPoly-Bergman Spaces and Two-dimensional SingularIntegral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

L. ZsidoWeak Mixing Properties of Vector Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 361

Editorial Preface

As this volume demonstrates, at roughly 100 years of age operator theory remainsa vibrant and exciting subject area with wide ranging applications. Many of thepapers found here expand on lectures given at the 15th International Workshop onOperator Theory and Its Applications, held at the University of Newcastle uponTyne from the 12th to the 16th of July 2004. The workshop was attended by closeto 150 mathematicians from throughout the world, and is the first IWOTA to beheld in the UK. Talks ranged over such subjects as operator spaces and their ap-plications, invariant subspaces, Kreın space operator theory and its applications,multivariate operator theory and operator model theory, applications of operatortheory to function theory, systems theory including inverse scattering, structuredmatrices, and spectral theory of non-selfadjoint operators, including pseudodiffer-ential and singular integral operators. These interests are reflected in this volume.As with all of the IWOTA proceedings published by Birkhauser Verlag, the pa-pers presented here have been refereed to the same high standards as those of thejournal Integral Equations and Operator Theory.

HILBERT

VON NEUMANN

C* ALGEBRAS

BANACH

H CONTROL

OPERATOR THEORY INTERPOLATION

KREIN

A few words about the above image which graced the workshop programmeand bag. In commuting between home in Hexham and work in Newcastle, I often

x Editorial Preface

travel by train. The journeys have resulted in a number of friendships, includingwith Chris Dorsett, who is a member of the Fine Arts department at NorthumbriaUniversity (also in Newcastle). A mathematics question led him to introduce mean article by the art critic and theorist, Rosalind Krauss titled “Sculpture in theExpanded Field”, which was first published in art journal October in 1978, and isnow recognized as a key work in contemporary art theory. To briefly summarizea portion of the thesis of her article, the term “sculpture” has been applied inthe 20th century to such a broad collection of art objects as to become essentiallymeaningless. This leads her to propose a refined classification built from the ideaof what sculpture is not (architecture, landscape) and the negation of these terms.The idea is encoded in a diagram much like the one given above, and is basedon a model of the Klein Viergruppe, also known as the Piaget group due to itsuse by the Swiss developmental psychologist Jean Piaget in the 1940’s to describethe development of logical reasoning in children. While the version pictured abovemakes a hash of the intended logic of the diagram (which would, for example,require that H∞ control be the negation of operator theory), it is nevertheless anhomage to Krauss and Chris Dorsett, indicates one of the unexpected ways thatart and mathematics come together, and encapsulates for me some of the salientfeatures of IWOTA.

Michael Dritschel

List of Participants

Alaa Abou-Hajar (University ofNewcastle)

Tsuyoshi Ando (HokkaidoUniversity) (Emeritus)

Jim Agler (University of California,San Diego)

Tuncay Aktosun (Mississippi StateUniversity)

Edin Alijagic (Delft University ofTechnology)

Calin-Grigore Ambrozie (Institute ofMathematics of the RomanianAcademy)

Robert Archer (University ofNewcastle)

Yury Arlinskii (East UkrainianNational University)

Damir Arov (Weizmann Institute)William Arveson (University of

California, Berkeley)Tomas Azizov (Voronezh State

University)Catalin Badea (University of Lille)Mihaly Bakonyi (Georgia State

University)Joseph Ball (Virginia Tech)

Oscar Bandtlow (University ofNottingham)

M. Amelia Bastos (InstitutoSuperior Tecnico)

Ismat Beg (Lahore University ofManagement Sciences)

Jussi Behrndt (TU Berlin)Sergio Bermudo (Universidad Pablo

de Olavide)Tirthankar Bhattacharyya (Indian

Institute of Science)Paul Binding (University of

Calgary)Debapriya Biswas (University of

Leeds)Danilo Blagojevic (Edinburgh

University)Albrecht Boettcher (Technical

University Chemnitz)Craig Borwick (University of

Newcastle upon Tyne)Lyonell Boulton (University of

Calgary)Maria Cristina Camara (Instituto

Superior Tecnico)Isabelle Chalendar (University of

Lyon)

xii List of Participants

Yemon Choi (University ofNewcastle)

Raul Curto (University of Iowa)Stefan Czerwik (Silesian University

of Technology)Volodymyr Derkach (Donetsk

National University)Patrick Dewilde (TU Delft)Aad Dijksma (Rijksuniversiteit

Groningen)Ronald Douglas (Texas A & M

University)Michael Dritschel (University of

Newcastle)Chen Dubi (Ben Gurion University

of the Negev)Harry Dym (The Weizmann

Institute of Science)Douglas Farenick (University of

Regina)Claudio Antonio Fernandes

(Faculdade de Ciencias eTecnologia, Almada)

Lawrence Fialkow (SUNY NewPaltz)

Alastair Gillespie (University ofEdinburgh)

Jim Gleason (University ofTennessee)

Israel Gohberg (Tel Aviv University)Zen Harper (University of Leeds)Mahir Hasanov (Istanbul Technical

University)Seppo Hassi (University of Vaasa)Munmun Hazarika (Tezpur

University)Georg Heinig (Kuwait University)William Helton (University of

California, San Diego)Sanne ter Horst (Vrije Universiteit

Amsterdam)Birgit Jacob (University of

Dortmund)

Martin Jones (University ofNewcastle)

Peter Junghanns (TechnicalUniversity Chemnitz)

Marinus A. Kaashoek (VrijeUniversiteit Amsterdam)

Michael Kaltenbaeck (TechnicalUniversity of Vienna)

Dmitry Kalyuzhnyi-Verbovetzkii(Ben Gurion University of theNegev)

Alexei Karlovich (Insituto SuperiorTecnico)

Yuri Karlovich (UniversidadAutonoma del Estado deMorelos)

Iztok Kavkler (Institute ofMathematics, Physics andMechanics, Ljubljana)

Laszlo Kerchy (University of Szeged)Alexander Kheifets (University of

Massachusetts Lowell)Vladimir V. Kisil (University of

Leeds)Edward Kissin (London

Metropolitan University)Martin Klaus (Virginia Tech)Artem Ivanovich Kozko (Moscow

State University)Ilya Krishtal (Washington

University)Matthias Langer (Technical

University of Vienna)Annemarie Luger (Technical

University of Vienna)Philip Maher (Middlesex University)Carmen H. Mancera (Universidad de

Sevilla)Stefania Marcantognini (Instituto

Venezolano de InvestigacionesCientıficas)

Laurent W. Marcoux (University ofWaterloo)

List of Participants xiii

John Maroulas (National TechnicalUniversity)

Rui Carlos Marreiros (Universidadedo Algarve)

Helena Mascarenhas (InstitutoSuperior Tecnico)

Vladimir Matsaev (Tel AvivUniversity)

John E. McCarthy (WashingtonUniversity)

Cornelis van der Mee (University ofCagliari)

Andrey Melnikov (Ben GurionUniversity of the Negev)

Christian Le Merdy (Universite deBesancon)

Blaz Mojskerc (University ofLjubljana)

Manfred Moller (University of theWitwatersrand)

Joachim Moussounda Mouanda(Newcastle University)

Ana Moura Santos (TechnicalUniversity of Lisbon)

Aissa Nasli Bakir (University ofChlef )

Zenonas Navickas (KaunasUniversity of Technology)

Ludmila Nikolskaia (Universite deBordeaux 1)

Nika Novak (University of Ljubljana)Mark Opmeer (Rijksuniversiteit

Groningen)Peter Otte (Ruhr-Universitat

Bochum)Jonathan R. Partington (University

of Leeds)Linda Patton (Cal Poly San Luis

Obispo)Jordi Pau (Universitat Autonoma de

Barcelona)Vern Paulsen (University of

Houston)

Alexander Sergeevich Pechentsov(Moscow State University)

Vladimir Peller (Michigan StateUniversity)

Luis Pessoa (Instituto SuperiorTecnico)

Sandra Pott (University of York)Geoffrey Price (United States Naval

Academy)Marek Ptak (University of

Agriculture, Krakow)Vladimir Rabinovich (National

Politechnic Institute of Mexico)Andre Ran (Vrije Universiteit

Amsterdam)Maria Christina Bernadette

Reurings (The College ofWilliam and Mary)

Stefan Richter (University ofTennessee)

Guyan Robertson (University ofNewcastle)

Helen Robinson (University of York)Steffen Roch (TU Darmstadt)Richard Rochberg (Washington

University)Leiba Rodman (College of William

and Mary)Karla Rost (Technical University

Chemnitz)James Rovnyak (University of

Virginia)Cora Sadosky (Howard University)Lev Sakhnovich (University of

Connecticut)Natasha Samko (University of

Algarve)Stefan Samko (University of

Algarve)Kristian Seip (Norwegian University

of Science and Technology)Eugene Shargorodsky (King’s

College London)

xiv List of Participants

Malcolm C. Smith (University ofCambridge)

Martin Paul Smith (University ofYork)

Rachael Caroline Smith (Universityof Leeds)

Henk de Snoo (RijksuniversiteitGroningen)

Michail Solomyak (The WeizmannInstitute of Science)

Olof Staffans (Abo Akademi)Jan Stochel (Uniwersytet

Jagiellonski)Vladimir Strauss (Simon Bolivar

University)Elizabeth Strouse (Universite de

Bordeaux 1)Franciszek Hugon Szafraniec

(Uniwersytet Jagiellonski)Richard M. Timoney (Trinity

College Dublin)Dan Timotin (Institute of

Mathematics of the RomanianAcademy)

Vadim Tkachenko (Ben GurionUniversity of the Negev)

Christiane Tretter (University ofBremen)

Nikolai Vasilevski (CINVESTAV delI.P.N., Mexico)

Luis Verde-Star (UniversidadAutonoma Metropolitana)

Sjoerd Verduyn Lunel (UniversiteitLeiden)

Victor Vinnikov (Ben GurionUniversity of the Negev)

Jani Virtanen (King’s CollegeLondon)

Dan Volok (Ben Gurion Universityof the Negev)

George Weiss (Imperial CollegeLondon)

Brett Wick (Brown University)Henrik Winkler (Rijksuniversiteit

Groningen)Harald Woracek (Technical

University of Vienna)Dmitry V. Yakubovich (Autonoma

University of Madrid)Nicholas J. Young (Newcastle

University)Joachim Peter Zacharias (University

of Nottingham)Laszlo Zsido (University of Rome)

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Talk Titles

p indicates plenary speakers, s semi-plenary speakers.

Jim Agler p

Function theory on analytic setsTuncay Aktosun

Inverse spectral-scattering problem with two sets of discrete spectral dataEdin Alijagic

Minimal factorizations of isometric operatorsCalin-Grigore Ambrozie p

Invariant subspaces and reflexivity results in Banach spacesRobert Archer

A commutant lifting theorem for n-tuples with regular dilationsYury Arlinskii

Realizations of H−2-perturbations of self-adjoint operatorsDamir Arov s

Bi-inner dilations and bi-stable passive scattering realizationsWilliam Arveson p

Free resolutions in multivariable operator theoryTomas Azizov

A uniformly bounded C0-semigroup of bi-contractionsCatalin Badea

Invertible extensions, growth conditions and the Cesaro operatorMihaly Bakonyi

Factorization of operator-valued functions on ordered groupsJoseph Ball

Realization and interpolation for Schur-Agler class functions on domains inCn with matrix polynomial defining function

Oscar BandtlowEstimates for norms of resolvents

M. Amelia BastosA symbol calculus for a C∗-algebra with piecewise oscillating coefficients

xx Talk Titles

Ismat BegConvergence of iterative algorithms to a common random fixed point

Jussi BehrndtSturm-Liouville operators with an indefinite weight and a λ-dependent bound-ary condition

Sergio BermudoA parametrization for the symbols of an operator of Hankel type

Paul BindingVariational principles for nondefinite eigenvalue problems

Albrecht Boettcher p

Asymptotical linear algebra illustrated with Toeplitz matricesLyonell Boulton

Effective approximation of eigenvalues in gaps of the essential spectrumMaria Cristina Camara

Polynomial almost periodic solutions for a class of oscillatory Riemann–Hilbert problems

Isabelle Chalendar p

Constrained approximation and invariant subspacesStefan Czerwik

Quadratic difference operators in Lp-spacesVolodymyr Derkach

Boundary relations and Weyl familiesPatrick Dewilde s

Matrix interpolation: the singular case revisitedAad Dijksma

Orthogonal polynomials, operator realizations, and the Schur transform forgeneralized Nevanlinna functions

Ronald DouglasEssentially reductive Hilbert modules

Chen DubiOn commuting operators solving Gleason’s problem

Harry DymA factorization theorem and its application to inverse problems

Douglas Farenick s

Some formulations of classical inequalities for singular values and traces ofoperators

Claudio Antonio FernandesSymbol calculi for nonlocal operator C∗-algebras with shifts

Lawrence FialkowA survey of moment problems on algebraic and semialgebraic sets

Talk Titles xxi

Alastair Gillespie p

An operator-valued principle with applicationsJim Gleason

Quasinormality of Toeplitz tuples with analytic symbolsIsrael Gohberg p

From infinite to finite dimensionsZen Harper

The discrete Weiss conjecture and applications in operator theoryMahir Hasanov

On perturbations of two parameter operator pencils of waveguide typeSeppo Hassi

On Kreın’s extension theory of nonnegative operatorsMunmun Hazarika

Recursively generated subnormal shiftsGeorg Heinig

Fast algorithms for Toeplitz and Toeplitz-plus-Hankel matrices with arbitraryrank profile

William Helton p

Noncommutative convexityBirgit Jacob s

What can semigroups do for nonlinear dissipative systems?Peter Junghanns

O(n log n)-algorithms for singular integral equations in the nonperiodic caseMarinus A. Kaashoek

Canonical systems with a pseudo-exponential potentialMichael Kaltenbaeck

Canonical differential equations of Hilbert-Schmidt typeDmitry Kalyuzhnyi-Verbovetzkii

On the intersection of null spaces for matrix substitutions in anon-commutative rational formal power series

Yuri KarlovichEssential spectra of pseudodifferential operators with symbols of limitedsmoothness

Alexei KarlovichBanach algebras of singular integral operators in generalized Lebesgue spaceswith variable exponent

Laszlo KerchySupercyclic representations

Alexander KheifetsOnce again on the commutant

xxii Talk Titles

Vladimir V. KisilSpectra of non-selfadjoint operators and group representations

Edward KissinLipschitz functions on Hermitian Banach ∗-algebras

Martin KlausRemarks on the eigenvalues of the Manakov system

Artem Ivanovich KozkoRegularized traces of singular differential operators of higher orders

Ilya KrishtalMethods of abstract harmonic analysis in the spectral theory of causal opera-tors

Matthias LangerVariational principles for eigenvalues of block operator matrices

Annemarie LugerRealizations for a generalized Nevanlinna function with its only generalizedpole not of positive type at ∞

Philip MaherOperator approximation via spectral approximation and differentiation

Laurent W. MarcouxAmenable operators on Hilbert space

Rui Carlos MarreirosOn the kernel of some one-dimensional singular integral operators with anon-Carleman shift

Helena MascarenhasConvolution type operators on cones and asymptotic properties

John E. McCarthy p

The Fantappie transformCornelis van der Mee

Approximation of solutions of Riccati equationsAndrey Melnikov

Overdetermined 2D systems which are time-invariant In one direction, andtheir transfer functions

Christian Le Merdy p

Semigroups and functional calculus on noncommutative Lp-spacesManfred Moller s

Problems in magnetohydrodynamics: The block matrix approachAna Moura Santos

Wave diffraction by a strip grating: the two-straight line approachAissa Nasli Bakir

Unitary nodes and contractions on Hilbert spaces

Talk Titles xxiii

Zenonas NavickasRepresenting solution of differential equations in terms of algebraic operators

Ludmila NikolskaiaToeplitz and Hankel matrices as Hadamard-Schur multipliers

Mark OpmeerIll-posed open dynamical systems

Peter OtteLiouville-type formulae for section determinants and their relation to Szegolimit theorems

Jonathan R. Partington p

Semigroups, diagonal systems, controllability and interpolationVern Paulsen p

Frames, graphs and erasuresAlexander Sergeevich Pechentsov

Asymptotic behavior of spectral function in singular Sturm–Liouville problemsVladimir Peller p

Extensions of the Koplienko-Neidhardt trace formulaeLuis Pessoa

Algebras generated by a finite number of poly and anti-poly Bergman projec-tions and by operators of multiplication by piecewise continuous functions

Sandra Pott s

Dyadic BMO and Besov spaces on the bidiskMarek Ptak

Reflexive finite dimensional subspaces are hyperreflexiveVladimir Rabinovich

Wiener algebras of operator-valued band dominated operators and its appli-cations

Andre Ran s

Inertia and hermitian block Toeplitz matricesStefan Richter

Analytic contractions, nontangential limits, and the index of invariant sub-spaces

Guyan RobertsonSingular masas of von Neumann algebras: examples from the geometry ofspaces of nonpositive curvature

Steffen RochFinite sections of band-dominated operators

Richard RochbergInterpolation sequences for Besov spaces of the complex ball

Leiba RodmanPolar decompositions in Krein spaces: An overview

xxiv Talk Titles

Karla RostInversion formulas for Toeplitz and Toeplitz-plus-Hankel matrices with sym-metries

James RovnyakOn indefinite cases of operator identities which arise in interpolation theory

Cora SadoskyBounded mean oscillation in product spaces

Lev SakhnovichIntegrable operators and canonical systems

Natasha SamkoSingular integral operators in weighted spaces of continuous functions with anon-equilibrated continuity modulus

Stefan SamkoMaximal and potential operators in function spaces with non-standard growth

Kristian SeipExtremal functions for kernels of Toeplitz operators

Eugene Shargorodsky s

Some open problems in the spectral theory of Toeplitz operatorsMalcolm C. Smith p

Foundations of systems theory and robust control: An operator theory per-spective

Martin Paul SmithSchatten class paraproducts and commutators with the Hilbert transform

Henk de SnooClosed semibounded forms and semibounded selfadjoint relations

Michail SolomyakOn a spectral properties of a family of differential operators

Olof StaffansState/signal representations of infinite-dimensional positive real relations

Jan StochelSelfadjointness of integral and matrix operators

Vladimir StraussOn an extension of a J-orthogonal pair of definite subspaces that are invariantwith respect to a projection family

Elizabeth StrouseProducts of Toeplitz operators

Franciszek Hugon SzafraniecA look at indefinite inner product dilations of forms

Richard M. TimoneyNorms of elementary operators

Talk Titles xxv

Dan TimotinA coupling argument for some lifting theorems

Vadim TkachenkoOn spectra of 1d periodic selfadjoint differential operators of order 2n

Christiane TretterFrom selfadjoint to nonselfadjoint variational principles

Nikolai VasilevskiCommutative C∗-algebras of Toeplitz operators on Bergman spaces

Luis Verde-StarVandermonde matrices associated with polynomial sequences of interpolatorytype sequences of interpolatory type

Sjoerd Verduyn Lunel s

New completeness results for classes of operatorsVictor Vinnikov

Scattering systems with several evolutions, related to the polydisc, and multi-dimensional conservative input/state/output systems

Dan VolokThe Schur class of operators on a homogeneous tree: coisometric realizations

George WeissStrong stabilization of distributed parameter systems by colocated feedback

Brett WickRemarks on Product VMO

Henrik WinklerPerturbations of selfadjoint relations with a spectral gap

Harald WoracekCanonical systems in the indefinite setting

Dmitry V. Yakubovich s

Linearly similar functional models in a domainJoachim Peter Zacharias

K-theoretical duality for higher rank Cuntz-Krieger algebrasLaszlo Zsido

Weak mixing properties for vector sequences

Operator Theory:Advances and Applications, Vol. 171, 1–16c© 2006 Birkhauser Verlag Basel/Switzerland

Inverse Scattering to Determinethe Shape of a Vocal Tract

Tuncay Aktosun

Abstract. The inverse scattering problem is reviewed for determining the crosssectional area of a human vocal tract. Various data sets are examined resultingfrom a unit-amplitude, monochromatic, sinusoidal volume velocity sent fromthe glottis towards the lips. In case of nonuniqueness from a given data set,additional information is indicated for the unique recovery.

Mathematics Subject Classification (2000). Primary 35R30; Secondary 34A5576Q05.

Keywords. Inverse scattering, speech acoustics, shape of vocal tract.

1. Introduction

A fundamental inverse problem related to human speech is [6, 7, 18, 19, 22] todetermine the cross sectional area of the human vocal tract from some data. Thevocal tract can be visualized as a tube of 14–20 cm in length, with a pair of lipsknown as vocal cords at the glottal end and with another pair of lips at the mouth.In this review paper, we consider various types of frequency-domain scattering dataresulting from a unit-amplitude, monochromatic, sinusoidal volume velocity inputat the glottis (the opening between the vocal cords), and we examine whethereach data set uniquely determines the vocal-tract area, or else, what additionalinformation may be needed for the unique recovery.

Human speech consists of phonemes; for example, the word “book” consistsof the three phonemes /b/, /u/, and /k/. The number of phonemes may vary fromone language to another, and in fact the exact number itself of phonemes in alanguage is usually a subject of debate. In some sense, this is the analog of thenumber of colors in a rainbow. In American English the number of phonemes is 36,39, 42, 45, or more or less, depending on the analyst and many other factors. The

The research leading to this article was supported in part by the National Science Foundationunder grant DMS-0204437 and the Department of Energy under grant DE-FG02-01ER45951.

2 T. Aktosun

phonemes can be sorted into two main groups as vowels and consonants. The vowelscan further be classified into monophthongs such as /e/ in “pet” and diphthongsuch as the middle sound in “boat.” The consonants can further be classified intoapproximants (also known as semivowels) such as /y/ in “yes,” fricatives such as/sh/ in “ship,” nasals such as /ng/ in “sing,” plosives such as /p/ in “put,” andaffricates such as /ch/ “church.”

The production of human speech occurs usually by inhaling air into the lungsand then sending it back through the vocal tract and out of the mouth. The airflow is partially controlled by the vocal cords, by the muscles surrounding the vocaltract, and by various articulators such as the tongue and jaw. As the air is pushedout of the mouth the pressure wave representing the sound is created. The effect ofarticulators in vowel production is less visible than in consonant production, andone can use compensatory articulation, especially in vowel production, by ignoringthe articulators and by producing a phoneme solely by controlling the shape of thevocal tract with the help of surrounding muscles. Such a technique is often usedby ventriloquists.

From a mathematical point of view, we can assume [18, 19] that phonemesare basic units of speech, each phoneme lasts about 10–20 msec, the shape of thevocal tract does not change in time during the production of each phoneme, thevocal tract is a right cylinder whose cross sectional area A varies along the distancex from the glottis. We let x = 0 correspond to the glottis and x = l to the lips.We can also assume that A is positive on (0, l) and that both A and its derivativeA′ are continuous on (0, l) and have finite limits at x = 0+ and x = l−. In fact,we will simply write A(0) for the glottal area and A(l) for the area of the openingat the lips because we will not use A or A′ when x /∈ [0, l].

Besides A(x), the primary quantities used in the vocal-tract acoustics arethe time t, the pressure p(x, t) representing the force per unit cross sectional areaexerted by the moving air molecules, the volume velocity v(x, t) representing theproduct of the cross sectional area and the average velocity of the air moleculescrossing that area, the air density µ (about 1.2 × 10−3 gm/cm3 at room tem-perature), and the speed of sound c (about 3.43 × 104 cm/sec in air at roomtemperature). In our analysis, we assume that the values of µ and c are alreadyknown and we start the time at t = 0. It is reasonable [18, 19] to assume thatthe propagation is lossless and planar and that the acoustics in the vocal tract isgoverned [6, 7, 18, 19, 22] by⎧⎨⎩A(x) px(x, t) + µ vt(x, t) = 0,

A(x) pt(x, t) + c2µ vx(x, t) = 0,(1.1)

where the subscripts x and t denote the respective partial derivatives.In order to recover A, we will consider various types of data for k ∈ R+

resulting from the glottal volume velocity

v(0, t) = eickt, t > 0. (1.2)

Inverse Scattering in a Vocal Tract 3

Note that the quantityck

2πis the frequency measured in Hertz. Informally, we will

refer to k as the frequency even though k is actually the angular wavenumber.The recovery of A can be analyzed either as an inverse spectral problem or as

an inverse scattering problem. When formulated as an inverse spectral problem,the boundary conditions are imposed both at the glottis and at the lips. Theimposition of the boundary conditions at both ends of the vocal tract results instanding waves whose frequencies form an infinite sequence. It is known [9, 11,14, 15, 17–19] that A can be recovered from a data set consisting of two infinitesequences. Such sequences can be chosen as the zeros and poles [15, 17] of theinput impedance or the poles and residues [11] of the input impedance.

If the recovery of A is formulated as an inverse scattering problem, a boundarycondition is imposed at only one end of the vocal tract – either at the glottis or atthe lips. The data can be acquired either at the same end or at the opposite end.We have a reflection problem if the data acquisition and the imposed boundarycondition occur at the same end of the vocal tract. If these occur at the oppositeends, we have a transmission problem. The inverse scattering problem may besolved either in the time domain or in the frequency domain, where the data set isa function of t in the former case and of k in the latter. The reader is referred to[4, 18–21, 23] for some approaches as time-domain reflection problems, to [16] foran approach as a time-domain transmission problem, and to [1] for an approachas a frequency-domain transmission problem.

The organization of our paper is as follows. In Section 2 we relate (1.1) tothe Schrodinger equation (2.3), introduce the selfadjoint boundary condition (2.7)involving cotα given in (2.8), and present the Jost solution f , the Jost functionFα, and the scattering coefficients T , L, and R. In Section 3 we introduce thenormalized radius η of the vocal tract, relate it to the regular solution ϕα to thehalf-line Schrodinger equation, and also express η in terms of the Jost solution andthe scattering coefficients. In Section 4 we present the expressions for the pressureand the volume velocity in the vocal tract in terms of the area, the Jost solution,and the Jost function; in that section we also introduce various data sets that willbe used in Section 6. In Section 5 we review the recovery of the potential in theSchrodinger equation and the boundary parameter cotα from the amplitude ofthe Jost function; for this purpose we outline the Gel’fand-Levitan method [3, 12,13, 10] and also the method of [13]. In Sections 6 we examine the recovery of thepotential, normalized radius of the tract, and the vocal-tract area from variousdata sets introduced in Section 4. Such data sets include the absolute values ofthe impedance at the lips and at the glottis, the absolute value of the pressure atthe lips, the absolute value of a Green’s function at the lips associated with (2.6),the reflectance at the glottis, and the absolute value of the transfer function fromthe glottis to the lips. Finally, in Section 7 we show that two data sets containingthe absolute value of the same transfer function but with different logarithmicderivatives of the area at the lips correspond to two distinct potentials as well astwo distinct normalized radii and two distinct areas for the vocal tract.

4 T. Aktosun

2. Schrodinger equation and Jost function

In this section we relate the acoustic system in (1.1) to the Schrodinger equation,present the selfadjoint boundary condition (2.7) identified by the logarithmic de-rivative of the area function at the glottis, and introduce the Jost solution, Jostfunction, and the scattering coefficients associated with the Schrodinger equation.

Letting

P (k, x) := p(x, t) e−ickt, V (k, x) := v(x, t) e−ickt, (2.1)

we can write (1.1) as{A(x)P ′(k, x) + icµk V (k, x) = 0,

c2µ V ′(k, x) + ick A(x)P (k, x) = 0,(2.2)

where the prime denotes the x-derivative. Eliminating V in (2.2), we get

[A(x)P ′(k, x)]′ + k2A(x)P (k, x) = 0, x ∈ (0, l),

or equivalentlyψ′′(k, x) + k2ψ(k, x) = Q(x)ψ(k, x), (2.3)

with

ψ(k, x) =:√

A(x) P (k, x), Q(x) :=[√

A(x)]′′√A(x)

. (2.4)

Alternatively, lettingΦ(x, t) :=

√A(x) p(x, t), (2.5)

we find that Φ satisfies the plasma-wave equation

Φxx(x, t) − 1c2

Φtt(x, t) = Q(x)Φ(x, t), x ∈ (0, l), t > 0. (2.6)

We can analyze the Schrodinger equation in (2.3) on the full line R by usingthe extension Q ≡ 0 for x < 0 and x > l. We can also analyze it on the halfline R+ by using the extension Q ≡ 0 for x > l and by imposing the selfadjointboundary condition

sin α · ϕ′(k, 0) + cosα · ϕ(k, 0) = 0, (2.7)

where

cotα := − A′(0)2 A(0)

= − [√

A(x)]′∣∣x=0√

A(0). (2.8)

As solutions to the half-line Schrodinger equation, we can consider the regularsolution ϕα satisfying the initial conditions

ϕα(k, 0) = 1, ϕ′α(k, 0) = − cotα, (2.9)

and the Jost solution f satisfying the asymptotic conditions

f(k, x) = eikx[1 + o(1)], f ′(k, x) = ik eikx[1 + o(1)], x → +∞.

Inverse Scattering in a Vocal Tract 5

Since Q ≡ 0 for x > l, we have

f(k, l) = eikl, f ′(k, l) = ik eikl. (2.10)

The Jost function Fα associated with the boundary condition (2.7) is defined [3,10, 12, 13] as

Fα(k) := −i[f ′(k, 0) + cotα · f(k, 0)], (2.11)

and it satisfies [3, 10, 12, 13]

Fα(k) = k + O(1), k → ∞ in C+, (2.12)

Fα(−k) = −Fα(k)∗, k ∈ R, (2.13)

where C+ is the upper half complex plane, C+ := C+∪R, and the asterisk denotescomplex conjugation.

Associated with the full-line Schrodinger equation, we have the transmissioncoefficient T , the left reflection coefficient L, and the right reflection coefficient Rthat can be obtained from the Jost solution f via

f(k, 0) =1 + L(k)

T (k), f ′(k, 0) = ik

1 − L(k)T (k)

, R(k) = −L(−k)T (k)T (−k)

. (2.14)

The scattering coefficients satisfy [2, 5, 12, 13]

T (−k) = T (k)∗, R(−k) = R(k)∗, L(−k) = L(k)∗, k ∈ R. (2.15)

In the inverse scattering problem of recovery of A, the bound states forthe Schrodinger equation do not arise. The absence of bound states for the full-line Schrodinger equation is equivalent for 1/T (k) to be nonzero on I+, whereI+ := i(0, +∞) is the positive imaginary axis in C+. For the half-line Schrodingerequation with the boundary condition (2.7) the absence of bound states is equiv-alent [3, 10, 12, 13] for Fα(k) to be nonzero on I+. It is known [3, 10, 12, 13] thateither Fα(0) �= 0 or Fα has a simple zero at k = 0; the former is known as thegeneric case and the latter as the exceptional case for the half-line Schrodingerequation. For the full-line Schrodinger equation we have T (0) = 0 generically andT (0) �= 0 in the exceptional case. The exceptional case corresponds to the thresholdat which the number of bound states may change by one under a small pertur-bation. In general, the full-line generic case and the half-line generic case do notoccur simultaneously because the former is solely determined by Q whereas thelatter jointly by Q and cotα.

3. Relative concavity, normalized radius, and area

In this section we introduce the normalized radius η of the vocal tract, relate it tothe regular solution to the Schrodinger equation, and present various expressionsfor it involving the Jost solution and the scattering coefficients.

6 T. Aktosun

Let r denote the radius of the cross section of the vocal tract so that A(x) =π [r(x)]2. Then, we can write the potential Q appearing in (2.4) as

Q(x) =r′′(x)r(x)

, x ∈ (0, l),

and hence we can refer to Q also as the relative concavity of the vocal tract. Define

η(x) :=

√A(x)√A(0)

, (3.1)

or equivalentlyA(x) = A(0) [η(x)]2, x ∈ (0, l). (3.2)

We can refer to η as the normalized radius of the vocal tract and η2 as the nor-malized area of the tract. From (2.4) we see that η satisfies

y′′ = Q(x) y, x ∈ (0, l),

with the initial conditions

η(0) = 1, η′(0) = − cotα.

A comparison with (2.9) shows that η is nothing but the zero-energy regular so-lution, i.e.,

η(x) = ϕα(0, x), x ∈ (0, l). (3.3)With the help of the expression [3, 10, 12, 13]

ϕα(k, x) =12k

[Fα(k) f(−k, x) − Fα(−k) f(k, x)] ,

we can write (3.3) as

η(x) = Fα(0) f(0, x) − Fα(0) f(0, x), x ∈ (0, l),

where an overdot indicates the k-derivative. We can also express η with the helpof the scattering coefficients. In the full-line generic case we get [1]

η(x) =

∣∣∣∣∣∣∣∣0 − i

2T (0) f(0, x) i f(0, x) − i

2R(0) f(0, x)

1 f(0, 0) 1

− cotα f ′(0, 0) 0

∣∣∣∣∣∣∣∣ , x ∈ (0, l),

and in the full-line exceptional case we have [1]

η(x) = f(0, x)

∣∣∣∣∣∣∣∣∣∣∣0 1

∫ x

0

dz

[f(0, z)]2

−1 f(0, 0) 0

cotα 01

f(0, 0)

∣∣∣∣∣∣∣∣∣∣∣, x ∈ (0, l). (3.4)

It is also possible to express η in other useful forms. Let g(k, x) denote thecorresponding Jost solution when we replace the zero fragment of Q for x ∈ (l, +∞)by another piece which is integrable, has a finite first moment, and does not yield

Inverse Scattering in a Vocal Tract 7

any bound states. Let τ(k), �(k), and ρ(k) be the corresponding transmission coeffi-cient, the left reflection coefficient, and the right reflection coefficient, respectively.In the generic case, i.e., when τ(0) = 0, we have [1]

η(x) =

∣∣∣∣∣∣∣∣0 − i

2τ(0) g(0, x) i g(0, x) − i

2ρ(0) g(0, x)

1 g(0, 0) 1

− cotα g′(0, 0) 0

∣∣∣∣∣∣∣∣ , x ∈ (0, l),

and in the exceptional case, i.e., when τ(0) �= 0, we have [1]

η(x) = g(0, x)

∣∣∣∣∣∣∣∣∣∣∣0 1

∫ x

0

dz

[g(0, z)]2

−1 g(0, 0) 0

cotα 01

g(0, 0)

∣∣∣∣∣∣∣∣∣∣∣, x ∈ (0, l).

4. Pressure and volume velocity in the vocal tract

In this section we present the expressions for the pressure and the volume velocityin the vocal tract in terms of the Jost function and the Jost solution. We alsointroduce various data sets associated with the values of the pressure and thevolume velocity at the glottal end of the vocal tract or at the lips.

From (1.2) and (2.1), we see that V (k, 0) = 1. Further, under the reasonableassumption that there is no reflected pressure wave at the mouth and all thepressure wave is transmitted out of the mouth, we get [1]

P (k, x) = − cµk f(−k, x)√A(0)

√A(x) Fα(−k)

, x ∈ (0, l), (4.1)

V (k, x) = − i√

A(x)√A(0)Fα(−k)

[f ′(−k, x) − A′(x)

2 A(x)f(−k, x)

], x ∈ (0, l), (4.2)

where we recall that f is the Jost solution to the Schrodinger equation and Fα isthe Jost function appearing in (2.11). Then, the pressure p(x, t) and the volumevelocity v(x, t) in the vocal tract are obtained by using (4.1) and (4.2) in (2.1).

The transfer function from the glottis to the point x is defined as the ratiov(x, t)/v(0, t), and in our case, as seen from (1.2) and (2.1), that transfer functionis nothing but V (k, x). In particular, the transfer function T(k, l) at the lips isobtained by using (2.10) in (4.2), and we have

T(k, l) =

√A(l) e−ikl√

A(0)Fα(−k)

[−k +

i

2A′(l)A(l)

],

|T(k, l)|2 =A(l)

A(0) |Fα(k)|2[k2 +

[A′(l)]2

4[A(l)]2

], k ∈ R. (4.3)