Lecture Notes in Mathematics - Springer978-3-540-37827-3/1.pdf · Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann Series: Institute of Mathematical Sciences. Madras

Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann Series: Institute of Mathematical Sciences. Madras Advisor: K R. Unni

399

Functional Analysis and its Applications International Conference, Madras, 1973

Edited by H. G. Garnir, K R. Unni and J. H. Wi lliamson

Springer-Verlag Berlin· Heidelberg· New York 1974

H. G. Garnir Universite de Liege/Belgium

K. R. Unni Matscience, Mad ras/lnd ia

J. H. Williamson University of York/ England

Library of Congress Cataloging in Publicalion Data

Main entry under title:

Functional analysis and i ts applications.

(Lecture notes in mathemat ics , 399) 1. Functional analysis --Congresses . I.

Henr j G., ed. II. Unni , K. R., 1933-III. Williamson, Jo~~ Hunter, 1926-IV. Series : Lecture notes in mathematics Q)\.3 .L28 no~ 399 . [ Q)\.320) 510 ' .8s [515 '. '7]

AMS Subject Classifications (1970): 46-XX

Garnir, ed.

ed . (Berlin) 399.

74-13998

ISBN 3-540-06869-4 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-06869-4 Springer-Verlag New York' Heidelberg· Berlin

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks.

Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.

© by Springer-Verlag Berlin' Heidelberg 1974.

Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

This volume is dedicated to Professor

ALLADI RAMAKRISHNAN, Founder-Director of MATSCIENCE,

The Institute of Mathematical Sciences, Madras,

who has completed twenty-five years of active

research in the realm of mathematical sciences.

PRE F ACE

This International Confer ence on Functional Analysis was

held at Madras from J anuary 1 to January 7, 1973 on the occasion

of the Eleventh Anniversary of Matscience, The Institute of Mathe

matical SCiences, Madras.

Functional analysis is not only a very important branch of

pure mathematics but perhaps the most fundamental for various

applications. It is also a subject in which the members of

Matscience have actively contributed.

In this conference, the emphasis was made on different aspects

of Functional AnalYSiS, namely Topological Vector Spaces, Harmonic

Analysis, Approximation Theory and Partial Differential EquatiOns.

This meeting was unique in the sense that it was the

biggest international conference in pure mathematics so far held

in India. There were more than 80 participants including 50

eminent mathematicians from more than 20 countries outSide India.

There were one hour invited lectures, half hour invited lectures

and ten minutes communications. In addition to the papers presen

ted at the conference, this volume also contains three papers

submitted in writing to the conference by the authors who could

not participate at the last minute.

This conference was inaugurated by the Patron of the

Institute, Hon'ble Sri C.Subramaniam, Minister of Science and

TeChnology, Government of India, who is also the founding father

of Matsclence. It is with pleasure that we record our sincere

gratitude to him and to the Chairman of the Board of Governors,

Hon'ble Thiru V.R.Nedunchezhiyan, Minister of Education, Govern

ment of Tamil Nadu, for their unfailing interest in the growth and

development of the Institute.

6

For the conduct of this conference, the Institute has

received full support and cooperation from the Government of Tamil

Nadu and the Government of India and the good will and cooperation

from the international mathematieal community and we wish to

express our warmest gratitude to them. Our thanks are due to the

citizens of Madras who have shown a keen interest in the programmes

of the Institute and in particular to Dr. Malcolm S.Adiseshiah,

former Acting Deputy Director General of the UNESCO, who presided

over the inaugural function.

Our particular thanks are due to Miss Vimala Walter who has

undertaken the tedious task of filling up all the mathematical

equations for the entire proceedings and to the staff of the

Institute namely Messrs R.Jayaraman, S.Krishnan, N.S.Sampath and

A.R.Balakrishnan for typing various manuscripts.

H. G.G.

K.R.U.

CONTENTS

Dedication . . . . . . . . . . . . . . . . . . . . 3

Preface • . . . . . . . . . . . . 5

Contents . . . . . . . . . . . . . 7

List of Participants •••••••••• 10

Lecture Programme • • . . . . . . . . . . . . 14

C.A.AKEMANN, Nonabelian Pontryagin Duality 18

R.A.ALO and Andre DE KORVIN, The Topological Dual, the Algebraic Dual and Radon Nikodym Derivatives 22

J.T.BURNHAM, Segal Algebras and Dense Ideals in Banach 33

Algebras

L.COLLATZ, Field Approximation and Free Approximation for Differential Equations • 59

J.G.DHOMBRES, Linear Interpolation and Linear Extension of Functions • • • 75

PH.A.DIONNE, Wave Front Sets and HyPoelliptic Operators 84

M.M.DZHRBASHYAN, Harmonic Analysis in the Complex Domain and Applications in the Theory of Analytical and Infinitely Differentiable Functions • • • • •• 94

LE BARON O.F&RGUSON,Mlintz-Szasz Theorem with Integral Coefficients I • • • • • • • • • • • • • • • • • • •• 119

J.P.FERRIER, Inductive Limits of Banach Spaces and Complex Analysis •• •.•••••..•••••••.••• 123

S.H.FESMIRE, Representation of Non-linear Operators with the Hammerstein Property • • • • • • • • • • • • • • •• 129

G.FREUD, On Polynomial Approximation with Respect to General Weights • • • • • • . . . . . . . . . . . . . . 149

D.GAIER, Determination of Conformal Modules of Ring Domains and Quadrilaterals • • • • • • • • 180

H.G.GARNIR, Solovay's Axiom and Functional Analysis. • 189

B.R.GELBAUM, Q-uniform Algebras and Operator Theory. • 205

A.GIROUX, On Polynomials with a Prescribed Zero. • • • •• 200

8

J. GOBERT, A Priori Inequalities for Systems of Partial Differential Equations •••••••• 0 0 • 0 • 0 212

R.R.GOLDBERG, Recent Results on Segal Algebras

K.N.GOWRISANKARAN, Measurability of Lattice Operators 1n a Cone . . . . . . . . . . . . . . . . . . .

220

230

p.HESS, On Some Non-linear Elliptic Boundary Value Problems 235

T . HUSAIN, I Quas icomplemented Banach Algebras • • • 248

S. IGARI, On the (LP ,LP) -4>1ul tipliers • • • • • • • • • • 254

JoP.KAHANE, Heredity in Metric Projections 258

G.N.KESHAVA MURTHY and K.R.UNNI, Multipliers on Weighted Spaces . . . . . . . . . . . . . . . . . . . .

L.D.KUDRYAVTSEV, On Properties of Traces of Functions belonging to Weight Spaces •• 0 • • • • • • •

P.LEONARD, Fundamental Solutions of Hyperbolic Differential

272

292

Equations ••••••••• •••••••••••• 313

F. T. MAKSUDQV and S.A. VELIEV, Non-self -conjugate Differential Dirac Operators Expansion in Eigenfunctions through the Whole Axis • • 0 • • 0 0 321

A.MALLIOS, Topological Algebras in Several Complex variable: 342

B.MONTADOR, Spectra of Compcsition Operators on C[O,l] 0 378

CoW.MULLINS, Linear Functionals on Vector Valued Kothe Spaces 380

BoSZ. -lIAGY, A General View on Unitary Dilations 382

E.T.POULSEN, Quantization in Hamiltonian Particle Mechanics 396

T.J .RIVLIN, The Lebesgue Constants for Polynomial Inter-polation . . . . . . . . . . . 422

K.SCHERER, Approximation Theorems for Polynomials Spline Operators • • 0 • • • 438

J • SCHMETS , Characterization of the Barrelled, d -Barrelled and ~-Barrelled Spaces of Continuous Functions ••• 468

I.J.SCHOENBERG, Cardinal Spline Interpolation and the Exponential Euler Splines ••••••••••••• 477

9

B. SENDOV, Approximation of Analytic Function in Hausdorff

Metric • • • • • • • • • • • • • 490

A.H.SIDDIQI, Invariant Means and Almost Convergence in

Non-archimedean Analysis • • • • • 501

K.R.UNNI, Parameasures and Multipliers of Segal Algebras 511

K.R.UNNI, Segal Algebras of Beurling Type. . . . . VIMALA WALTER, Splines in Hilbert Spaces

S. G. WAYMENT, A Survey of v -integral Representation Theory

for operators on Function Spaces including the Topo-

529

538

logical Vector Space Setting • • • • • • • •• 561

J .H. rlILLIAMSON, Isotone Measures 1948-1973 573

Unsolved Problems . . . . . . . . . . . 584

C.A.AKEMANN

W.A.A.L SALAAM

R.A.ALO<*)

C.BERGE

J. T. BURNHAM( *)

L.COLLATZ

T.R. DHANAPALAN

J. G.DHOMBRES

PH.A.DIONNE

M.M.DZHRBASHYAN

LE BARON O.FERGUSON

J.p.FERRIER

S.H.FESMIRE( *)

G.FREUD

D.GAIER

H.G.GARNIR

P.K.GEETHA

LIST OF PARTICIPANTS

University of California, Santa Barbara, California 93106, U.S.A. and University of Copenhagen, Denmark.

University of Alberta, Edmonton, Canada.

Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, U.S.A.

C.E.N., Saclay, Paris, France.

University of York, Heslington, York, England and University of Iowa, Iowa City 52240, U.S.A..

Institut fUr angewandte Mathematik, Universitat Hamburg, Hamburg, West Germany.

Annamalai University, Annamalainagar, India

Universite de Nantes, 44 Nantes, France.

University of Auckland, Auckland, New Zealand.

Institute of Mathematics, Armenian Academy of SCiences, Erevan, U.S.S.R.

University of California, Riverside, California 92502, U.S.A.

Universite de Nancy, Nancy, France.

Carnegie-Mellon University, Pittsburgh, Pennsylvania 15232, U.S.A.

Mathematical Institute of the Hungarian Academy of Sciences, Budapest, Hungary.

Justus Liebig Universitat, Giessen, West Germany. Institut de Mathematique, Universite de Liege, 4000 Li~ge, Belgium.

Matscience, Madras, India.

B .R. GELBAUM

A.GIROUX

J.GOBERT

R.R.GOLDBERG

K. GOPALASAMY

J.P.GOSSEZ

X.N.GOWRISANKARAN

A.K.GUPTA

s.c. GUPTA

A.GUSSEINOV

p.HESS

T.HUSAIN

S.IGARI

R.JAGANNATliA.N

H.JOHNEN

J.P.:KA.HiUlE

G.K.KALISCH

N.KASTHURI

G.N.KESHAVA MURTHY

KRISHNASWAMI ALLAnI

11

SUNY at Buffalo, Buffalo, New York 14214, U.S.A.

Universite de Montreal, Montreal, Canada.

Institut de Math~matique, Universite de Liege, 4000 Liege, Belgium.

University of Iowa, IO~Ta City, Iowa 52240,

U.S:.A.

The Flinders University of South Australia, Bedford Park, South Australia 5)42.

Free University of BrusselS, Brussels, Belgium.

McGill University, Montreal, Canada.

Indian Institute of Technology, Kanpur, India.

Aligarh Muslim University, Aligarh, India.

Mathematical Institute of Academy of Sciences of Azerbaidjan, Baku, U. S.S. R.

University of Queensland, St. Lucia, Brisbane, Australia.

Universitiit ZUriCh, 8032 ZUrich, Switzerland.

McMaster University, Hamilton, Ontario, Canada.

Tohoku University, Sendal, Japan.

Matsclence, Madras, India.

Techn. Hochschule Aachen, 51 Aachen, West Germany.

Unlversite de PariS, Orsay, France.

University of California, IrVine, California 92664, U.S.A.



Vivekananda College, Madras, India.

L.D.KUDRYlVTSEV

P.LEONARD

F.E.MACSDDOV

A.MALLIOS

S.MANI

K.H.MARIWALLA

B.MONTADOR

C.W.MULLINS

G.S.N.MURTHY

B.SZ.-NAGY

L.NIRENBERG

F.J.NORONHA

D.V.PAI

A.PELCZYNSKI

S.T.POULSEN

Y.S.PRAHLAD

V .RADHAKR ISHNAN

ALLAnI RAMAKRISHNAN

N.R.RA.NGANATBAN

T.J • RIVL IN

T. SATYANARAYANA

K,SCHERER

12

Steklov Mathematical Institute, Moscow, U.S.S.R.

Institut de Mathematique, Universite de Liege, 4000 Liege, Belgium.

Mathemat i cal Ins titut e of the Academy of Sciences of Azerbaid j an, Baku, U. S . S .R.

Mathematical Institute, University of Athens, Athens, Greece.



Universite de Sherbrooke, Sherbrooke, Canada.

Pahlavi University, Shiraz, Iran.


UniverSity of S7.eged,Szeged, Hungary.

Courant Institute of Mathematical SCiences, New York University, New York 10012, U.S.A.

UniverSity of Bangalore, Bangalore, India.

Indian Institute of Technology, Bombay, India.

Institute of Mathematics, PoliSh Academy of SCiences, Warsaw, Poland.

Aarhus University, Aarhus, Denmark.





Thomas J.Watson Research Centre, York Town Heights, New York 10598, U.S.A.


Techn. Hochschule Aachen, 51 Aachen, West Germany.

J.SCHMETS

I.J.SCHOENBERG

B. SENDOV

R.N.SEN

A.H.SIDDI QI

K.R.SONAVANE

R.SRIDHAR

K.SRINIVASA RAO

R.P.SRIVATSAV

V.SUBBA RAO

M.R.SUBRAHMANYA

A. R. TEKUMALLA

K.R.UNNI

A.VIJAYAKUMAR

VlMALA WAL TEa

S.G.WAYMENT

J.H.WILLIAMSON

B.H.YOON

13

Institut de Mathematique, Universite de Liege, 4000 Liege, Belgium.

Mathematics Research Centre, University of Wisconsin, Madison, Wisconsin 53705, U.S.A.

University of Sofia, Sofia 26, Bulgaria.

University of Alberta, Edmonton, Canada.

Kalyani University, West Bengal, India.

University of Tabriz, Iran.

Shivaji University, Kolhapur, India.



SUNY at Stony Brooke, Stony Brooke, New York, U.S.A.

Indian Institute of Technology, Madras, India .


Matscience, Madras, India •




Utah State University, Logan, Utah" U.S.A.

University of York, Hesllngton, York, England.

Sogang University, Seoul, Korea.

(*)indicates persons whose papers are included in this volume, but could not participate.

10.00

16.00

FIRST SESSION

9.00 9.55

9.50 - 10.45

SEeo ND SESS 10 N

Section A.

1l.15

11.45 12.15

- 11.40

- 12.10 - 12.40

Section B.

LECTURE PROGRAMME

MondaY, 1st January, 1973

Registration of the participants. The registration continuous for the whole day.

Inanguration of the conference.

Tuesday, 2nd January, 1973

Chairman: A11adi Ramakr ishnan

I.J. Schoenberg, 'On cardinal spline interpolation', B.R.Gelbaum, 'Operator theory and Q-uniform Banach spaces',

Chairman : T. Husain

J.Schmets, 'Generalization of the theorem of Nachbin-Shirota about the spaces of cont i nuous fune tions ' • C. A.AJremarm, ' Non-abelian Pontryagin Duality'. A.Pel ezynskl, 'Subspaces of L -spaces'. p

Chairman 1 J.Gobert

11.15 - 11.40 K.Scherer, 'Approximation on compact homogeneous spaces '.

11.45 - 12.10 B.Ferguson, 'MUutz-Szasz theorem with integral coefficients "

12.15 - 12.40 Ph.A.Dionne, 'Wave front sets and hypoe1lipticity'.

THIRD SESSION

15.10 - 16.00

16.30 - 17.20

Chairman : B.R.Gel baum

A.Mallios, 'Topological algebras and several complex variables '. L.D.Kudryavtsev, 'On properties of traces of functions belonging to weight spaces' .

15

Wednesday, 3rd JanuarY 1973

FOURTH SESSION

C.hairman : J.P.Kahane

9.00 9.50 H.G.Garnir, 'The axiom of Solovay and functional analysis'.

9.55 - 10.45 B.Sz.-Nagy, 'A general view on the theory of uni tary dilations of operators',

FIFTH SESSION Section A.

11.15 11.45

12.15

12.40

- 11.40 - 12.10

12.40

- 12.50

Section B.

Chairman : S.Igari

T. Husain, 'QUasicomplemented Banach spaces', S.G.Wayment, 'v-integrals in topological vector spaces '. A.H.Siddiqi~ 'Invariant means and convergence in non-Archimedian analysis'. G.N.Keshava Murthy, 'On a class of functions satisfying Lipschitz condition'.

Chairman : P.Hess

11.15 - 11.40 J.P.Gossez, 'Non-linear elliptic boundary value problem in Orlicz - Sobolev spaces',

11.45 - 12.10 J,Gobert, 'A priori inequalities for systems of partial differential equations'.

12.15 - 12.40 G.Freud, 'On weighted polynomial approximation',

12.40 - 12.50 A.Sharma, 'Averaging interpolation',

SIXTH SESSION

Chairman: B,Sz.- Nagy

15.10 - 16.00 L. Nirenberg, 'Remarks on linear partial differential equations' •

16.30 - 17.20 D.Gaier, 'Approximation of modules of quadrilaterals and of ring domains'.

Thursday, 4th January 1973

SEVENTH SESSION C.hairman : L.D.Kudryavtsev

9.00 9.50 J.P.Kahane, 'Some problems in metric projection'.

9.55 - 10.45 Bl.Sendov, 'Some problems in Hausdorff approximation'. 11.15 - 12.05 L.Collatz, 'Field approximation in partial differ

ential equations'.

16

EIGHTH SESSION

Chairman : A.Mallios

12.15 - 13.05 J.H.Williamson, 'Isotone measures'.

13.10 New and unsolved problems.

Afternoon: Excurs ion

Friday, 5th January 1973

NINTH SESSION

Chairman : H.G.Garnir

9.00 - 9.50 E.T.Poulsen, 'Quantization in Hamiltonian Particle mechanics' •

9.55 - 10.45 R.R.Goldberg, 'Segal algebras'.

TENTH SESSION

Se.ction A. Chairman : R.R.Goldberg

11.15

11.45

12.15

12.40

- 11.40 J.G.Dhombres, 'Linear interpolation of functions on a compact space'.

- 12.10 G.N.Keshava Murthy, 'Multipliers on weighted spaces' - 12.40 J.P.Ferrier, 'Inductive limits of Banach spaces

and complex analysis'. - 12.50 M.Montad0r, 'The spectrum of composition operators

in C [O,lJ '. Section B. Chairman .: G.Freud

Ln .• IS

11.45

12.15

12.40

- 11.40 P.T,GODard? 'Fundamental solutions of hyperbolic differentlal equations',

- 12.10 H.Johnen, 'Approximation by splines'. - 12.40 Vimala Walter, 'Splines in Hilbert spaces'. - 12.50 A.Geroux, 'Polynomials with a prescribed Zero',

ELEVENTH SESSION Chairman L.Co11atz

15.10 - 16.00 G.K.Ka1isch, 'On unitary equivalence classes of certain normal operators and their unitary invariants' •

16.30 - 17.20 P.Hess, 'Non Coercive non linear boundary value problems and homotopY arguments'.

17

Saturday, 6th January 1973

TWEELFTH SESSION

Chairman E.T.Poulsen

9.00 9.50 T.J.Rivlin, 'The Lebesgue constant for polynomi al interpolation' •

9.55 - 10.45 S.Igari, 'On the (LP, LP) multipliers'.

THIRTEENTH SESSION

C.hairman T.J.Rivlin

11.15 - 12.05 M.M.Dzhrbashyan, 'Harmonic analysis in the complex domain and its application in the theory of infinitely differentiable functions'.

12.10 - 13.00 K.R. Unni, 'Hul tipliers of Segal algebras'.

Afternoon: Excursion

Sunday, 7th January 1973

FOURTEENTH SESSION

Section A.

10.00 - 10.25

10.30 - 10.55

11.25 - H.35

11.35 - 1l.45

11.45 - H.55

12.00 - 12.25

Section B.

10.00 - 10.25

10.30 - 10.55 11.00 - 1l.45

11.45 - 12.00

C.hairman J.H.Williamson

K.N.Gowrisankaran, 'Heasurability of lattice of operations on a cone and an application to potential theory'. R.P.Srivatsav, 'A Hilbert space teChnique for the solution of dual integral equations and dual series equations'. T.R.Dhanapalan, 'I4atrix Transformations of spaces of sequences in a Banach algebra'. K.R.Sonavane, 'Generalized functions depending on a parameter'. C. W.Mullins, 'Linear functionals on vector valued Kothe spaces'. F.Macsudov.

Chairman K.R.Unni

D.V.Pai, 'Approximation of vector valued functions'. M.R.Subrahmanya, 'On simultaneous best approximation '. R.N.Sen, 'On the iterative process of solving nonlinear operator equations'. P.K.Geetha, 'On Bernstein approxima tion problem'.

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Lecture Notes in Mathematics - Springer978-3-540-37827-3/1.pdf · Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann Series: Institute of Mathematical Sciences. Madras