Progress in Approximation Theory and Applicable Complex Analysis: In Memory of Q.I. Rahman

541 Pages·2017·6.36 MB·English

by  Narendra Kumar Govil

Springer Optimization and Its Applications 117
Narendra Kumar Govil
Ram Mohapatra
Mohammed A. Qazi
Gerhard Schmeisser Editors
Progress in
Approximation
Theory and
Applicable
Complex Analysis
In Memory of Q.I. Rahman
Springer Optimization and Its Applications
VOLUME117
ManagingEditor
PanosM.Pardalos(UniversityofFlorida)
Editor–CombinatorialOptimization
Ding-ZhuDu(UniversityofTexasatDallas)
AdvisoryBoard
J.Birge(UniversityofChicago)
C.A.Floudas(TexasA&MUniversity)
F.Giannessi(UniversityofPisa)
H.D.Sherali(VirginiaPolytechnicandStateUniversity)
T.Terlaky(LehighUniversity)
Y.Ye(StanfordUniversity)
AimsandScope
Optimization has been expanding in all directions at an astonishing rate
during the last few decades. New algorithmic and theoretical techniques
havebeendeveloped,thediffusionintootherdisciplineshasproceededata
rapidpace,andourknowledgeofallaspectsofthefieldhasgrownevenmore
profound.Atthesametime,oneofthemoststrikingtrendsinoptimization
is the constantly increasing emphasis on the interdisciplinary nature of the
field.Optimizationhasbeenabasictoolinallareasofappliedmathematics,
engineering,medicine,economics,andothersciences.
The series Springer Optimization and Its Applications publishes under-
graduate and graduate textbooks, monographs and state-of-the-art exposi-
tory work that focus on algorithms for solving optimization problems and
alsostudyapplicationsinvolvingsuchproblems.Someofthetopicscovered
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multi-objectiveprogramming,descriptionofsoftwarepackages,approxima-
tiontechniquesandheuristicapproaches.
Moreinformationaboutthisseriesathttp://www.springer.com/series/7393
Narendra Kumar Govil • Ram Mohapatra
Mohammed A. Qazi • Gerhard Schmeisser
Editors
Progress in Approximation
Theory and Applicable
Complex Analysis
In Memory of Q.I. Rahman
123
Editors
NarendraKumarGovil RamMohapatra
DepartmentofMathematics DepartmentofMathematics
andStatistics UniversityofCentralFlorida
AuburnUniversity Orlando,FL,USA
Auburn,AL,USA
GerhardSchmeisser
MohammedA.Qazi DepartmentofMathematics
DepartmentofMathematics UniversityofErlangen-Nuremberg
TuskegeeUniversity Erlangen,Germany
Tuskegee,AL,USA
ISSN1931-6828 ISSN1931-6836 (electronic)
SpringerOptimizationandItsApplications
ISBN978-3-319-49240-7 ISBN978-3-319-49242-1 (eBook)
DOI10.1007/978-3-319-49242-1
LibraryofCongressControlNumber:2017930625
Mathematics Subject Classification (2010): 11Y35, 26A33, 26D15, 30A10, 30B10, 30C10, 30C15,
30C35,30C45,30C80,30D35,30E10,30H05,31A10,33C45,33C47,41A10,41A17,41A20,41A21,
41A25,41A30,41A40,41A50,41A52,41A55,41A80,42A05,42A15,42A82,42C05,42C10,42C15,
46C05,47A07,49J40,65B10,65B15,65D30,65D32,65N30,90C33.
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Foreword
QaziIbadurRahman(QIR)
Q.I.RahmanwaswithDickAskeyoneoftheworld’sgreatestexpertsonestimates
in two important areas of complex analysis: functions of exponential type and
polynomials.
The number and variety of papers in this book show how great his influence
has been in this and other areas. His book with Gerhard Schmeisser The Analytic
Theory of Polynomials remains the fundamental textbook for anyone studying the
latterarea.Hisdistinctionhasbeenrecognizedbyhisbeingonlythe21stpersonand
thesecondmathematiciantobeawardedanhonorarydoctoratefromtheUniversity
MariaCurie-SkłodowskainLublin,Poland.ThefirstawardeewasIrèneCurie.
Letmementionsomeofhismostmemorableresults.
Theorem1(withB.D.BojanovandJ.SzynalinMathZ.190(1985),281–285).
Letf.z/beapolynomialofdegreen,allofwhosezerosz(cid:2) lieintheclosedunit
disk.Theneachdisk
v
vi Foreword
jz(cid:2)z(cid:2)j(cid:3)21=n (1)
contains at least one zero of f0.z/. This is still the strongest result obtained for
generalntowardSendov’sconjecturethattheright-handsidecanbereplacedby1
in(1).
TwofurtherstrikingresultsarecontainedinhispaperinTrans.Amer.Math.Soc.
163(1972),447–455.
Theorem 2. Suppose that p .z/ is a polynomial of degree n. Then if jp .x/j (cid:3)
n n
.1(cid:2)x2/1=2for(cid:2)1<x<1,then
jp0.x/j(cid:3)2.n(cid:2)1/ for(cid:2)1(cid:3)x(cid:3)1:
If jp .x/j(cid:3)jxj,(cid:2)1(cid:3)x(cid:3)1,then
n
jp0.x/j(cid:3).n(cid:2)1/2C1 for(cid:2)1(cid:3)x(cid:3)1:
Boththeseresultsaresharp.
QaziRahmanwaswithusatImperialCollegefrom1959to1961andwasamost
valuable member of our group. Other members were Jim Clunie, Noel Baker, and
ThomasKovari.
Laterin1965whenhebecameprofessorattheUniversitédeMontréal,heinvited
severalofustheretovisithimonanumberofoccasions.HewasadevoutMuslim
andsowerehiswifeFatimaandhissonsAmerandMujtaba.Ithasbeenrightlysaid
thatthefamilywhopraystogether,staystogether.
I particularly remember his kindness to fellow Muslims, my wife Waficka and
myself.Heevenofferedtomoveoutofhishomesowecouldstayinit.Iamdeeply
gratefulforhisfriendshipandlovethroughouthislifeandthatofFatima,Amer,and
Mujtaba.
This book, which follows, is dedicated to Q.I. Rahman, who will long be
rememberedforhismathematics,hisenthusiasmanddedicationfordoingresearch
inmathematics,andfortheoverwhelmingkindnessandunderstandingheshowered
onallwhocameincontactwithhim.
London,UK W.K.Hayman,F.R.S.
Preface
This volume is a collection of chapters dedicated to Professor Q. I. Rahman who
passed away on July 21, 2013, in Montreal, Canada. Professor Rahman was a
leading mathematician whose research spanned several areas of approximation
theory and classical analysis, including complex analysis. Professor Rahman was
viewedasaworldexpertintheanalytictheoryofpolynomialsandentirefunctions
ofexponentialtypebyhiscollaboratorsandmanyothercolleagues.
We invited outstanding mathematicians, friends, and collaborators of Professor
Rahmantosubmitchapterstobeincludedinthisvolume.Thiscollectioncontains
originalresearcharticlesandcomprehensivesurveycontributionsby36mathemati-
ciansfrom18countries.Allthechapterswererefereed.Wehopethatthechapters
willinterestgraduatestudentsandresearchersinanalysisandapproximationtheory.
Professor Walter K. Hayman, F.R.S., who was one of Professor Rahman’s
teachers,haspreparedthemonograph’sForeword.Weareextremelygratefultohim
forthetimeandeffortthathehasdevotedtothewritingofthispiece.
Thechaptersofthemonographaregroupedbyfourthemeswhichreflectsome
ofProfessorRahman’sareasofresearch.ThefirstthemeisPolynomials.Itincludes
inequalities for polynomials and rational functions, orthogonal polynomials and
locationofzeros,andcomprisesthechaptersentitled“OntheL2MarkovInequality
with Laguerre Weight”, “Markov-Type Inequalities for Products of Müntz Poly-
nomials Revisited”, “On Bernstein-Type Inequalities for the Polar Derivative of a
Polynomial”,“OnTwoInequalitiesforPolynomialsintheUnitDisk”,“Inequalities
for Integral Norms of Polynomials via Multipliers”, “Some Rational Inequalities
Inspired by Rahman’s Research”, “On an Asymptotic Equality for Reproducing
Kernels and Sums of Squares of Orthonormal Polynomials” and “Two Walsh-
Type Theorems for the Solutions of Multi-Affine Symmetric Polynomials”. The
second theme is Inequalities and Extremal Problems, where functions other than
polynomials are considered. This theme consists of chapters entitled “Vector
InequalitiesforaProjectioninHilbertSpacesandApplications”,“AHalf-Discrete
Hardy-Hilbert-TypeInequalitywithaBestPossibleConstantFactorRelatedtothe
Hurwitz Zeta Function”, “Quantum Integral Inequalities for Generalized Convex
Functions”, “Quantum Integral Inequalities for Generalized Preinvex Functions”
vii
viii Preface
and “On the Bohr Inequality”. The third theme is Approximation of Functions,
the approximants being polynomials, rational functions and other types of func-
tions; see Chapters entitled “Bernstein-Type Polynomials on Several Intervals”,
“Best Approximation by Logarithmically Concave Classes of Functions”, “Local
ApproximationUsingHermiteFunctions”,“ApproximatingtheRiemannZetaand
RelatedFunctions”,“OverconvergenceofRationalApproximantsofMeromorphic
Functions” and “Approximation by Bernstein-Faber-Walsh and Szász-Mirakjan-
Faber-WalshOperatorsinMultiplyConnectedCompactSetsofC”.Thelasttheme
is Quadrature, Cubature and Applications. It comprises three chapters, including
a posthumous article of Professor Rahman co-authored by one of the editors of
this book. This theme includes chapters entitled “Summation Formulas of Euler-
Maclaurin and Abel-Plana: Old and New Results and Applications”, “A New
Approach to Positivity and Monotonicity for the Trapezoidal Method and Related
QuadratureMethods”and“AUnifiedandGeneralFrameworkforEnrichingFinite
ElementApproximations”.
Inthefirstchapter,theauthorsNikolovandShadrinhaveconsideredL2 Markov
inequalitywithLaguerreweightoverasemi-infiniteintervaloftherealline.They
havealsoobtainedanasymptoticvalueoftheconstantintheirinequality.
In the chapter by Erdélyi, new Markov-type inequalities for products of Müntz
polynomials have been proved. These results extend some of the earlier contribu-
tionsoftheauthorandanswersomequestionsposedbyThomasBloom.
Govil and Kumar in their survey article mention in a chronological manner
Bernstein-type inequalities for polar derivatives of a polynomial. This chapter
providesacomprehensiveaccountofresultsonpolarderivatives.
FournierandRuscheweyhconsidertwoverydifferentgeneralizationsandrefine-
mentsofBernstein’sinequalityforpolynomialsthathavebeenobtainedmorethan
30yearsago.Here,theauthorsshowthatoneoftheseinequalitiesimpliestheother.
Theyalsostudythecasesofequality.
Pritskerconsidersawiderangeofpolynomialinequalitiesfornormsdefinedby
contour and area integrals over the unit disk in the complex plane. He has also
provedinequalitiesusingtheSchur-Szego˝ composition.
ThechapterbyLi,Mohapatra,andRanasingheisconcernedwithsomerational
inequalities inspired by Rahman’s research. The results include Bernstein-type
inequalitiesforrationalfunctionswithprescribedpolesandprescribedzeros.
IgnjatovicandLubinskyinvestigateanasymptoticequalityforreproducingker-
nelsandsumsofsquaresoforthonormalpolynomials.Theseresultsaremotivated
by the recent work of Ignjatovic on orthonormal polynomials associated with a
symmetric measure with unbounded support and satisfying a recurrence relation.
The authors have studied the case of even exponential weights and weights on a
finiteinterval.
In the chapter by B. Sendov and H. Sendov, the authors have considered two
Walsh-typetheoremsforthesolutionofmulti-affinesymmetricpolynomials.These
results can be considered as extensions of the Grace-Walsh-Szego˝ coincidence
theorem.
Preface ix
InthechapterbyDragomir,somevectorinequalitiesrelatedtothoseofSchwarz
and Buzano are established. Also, inequalities involving the numerical range and
thenumericalradiusfortwoboundedoperatorsareobtained.
RassiasandYanghaveusedmethodsofweightfunctionstoobtainahalf-discrete
Hardy-Hilbert-type inequality with a best constant related to the Hurwitz-Zeta
function.Equivalentforms,normedoperatorexpressions,theirreverses,andsome
specialcasesarealsoconsidered.
InthechapterbyM.Noor,K.Noor,andAwan,theauthorshaveconsideredand
generalizedconvexfunctionsinvolvingtwoarbitraryfunctionsandestablishedsome
new quantum integral inequalities for the generalized convex functions. Besides,
severalspecialcasesofinteresthavebeenmentionedascorollaries.
M.Noor,Rassias,K.Noor,andAwanhaveconsideredquantumintegralinequal-
ities involving generalized preinvex functions. They give an account of quantum
integral inequalities and in a certain limiting case use these inequalities to obtain
many well-known results as special cases. The contents of this chapter are related
tothatofthepreviouschapter.
In the chapter by Abu Muhanna, Ali, and Ponnusamy, the Bohr inequality is
considered.Thissurveyarticleconsidersrecentadvancesandgeneralizationsofthe
Bohr inequality in the unit disk of the complex plane. Among other things, they
havediscussedtheBohrradiusforharmonicandstarlikelogharmonicmappingsin
theunitdisk.
InthechapterbySzabados,Bernstein-typepolynomialsforasetJ ofsfinitely
s
manyintervalshavebeenconsidered.Onsuchsets,approximatingoperatorsresem-
bling Bernstein polynomials have been defined, and their interpolation properties
andrateofconvergenceareobtained.
The chapter by Dryanov contains results on best approximation by a class of
logarithmically concave functions. Exact values of best approximations are found
fortwospecificcases.
ThechapterbyMhaskarconsiderslocalapproximationusingHermitefunctions.
Hedevelopsawavelet-likerepresentationinL .R/wherethelocalbehaviorofthe
p
termscharacterizesthelocalsmoothnessofthetargetfunction.Hegivesnewproofs
forthelocalizationofcertainkernelsaswellasfortheMarkov-Bernsteininequality.
Stenger considers a function G which has the same zeros as the well-known
Riemannzetafunctioninthecriticalstrip.Forstudyingitsbehaviorforintermediate
valuesofz,heusesFourierseriesandderivesanasymptoticapproximationforlarge
valuesofz.
The chapter by Blatt deals with overconvergence of rational approximants of
meromorphic functions. It contains results on the degree of convergence and
distribution of zeros of the rational approximants. In addition, well-known results
onpolynomialapproximationofholomorphicfunctionsaregeneralized.
ThechapterbyGalisconcernedwithapproximationbyBernstein-Faber-Walsh
and Szász-Mirakjan-Faber-Walsh operators in multiply connected compact sets
of the complex plane. These results are generalizations of earlier results of the
authoronq-Bernstein-FaberpolynomialsandSzász-Faber-typeoperatorsinsimply
x Preface
connectedcompactsetsinC.Thisstudyleadstoaconjectureconcerningtheuseof
truncatedclassicalSzász-Mirakjanoperatorsinweightedapproximation.
Milovanovic´ discusses old and new results on summation formulas of Euler-
Maclaurin and Abel-Plana. He has shown connections between Euler-Maclaurin
formulaandbasicquadraturerulesofNewton-Cotes-typeaswellasthecomposite
Gauss-LegendreruleanditsLobatomodifications.Summationformulassuchasthe
midpointsummationformula,theBinetformula,andtheLindelöfformulaarealso
extendedandanalyzed.
The chapter by Rahman and Schmeisser provides a new approach to positivity
and monotonicity for quadrature methods. In all of the known results, sign condi-
tionsonsomederivativesofthegivenfunctionarerequired.Theauthorsproposea
newapproachbasedonFourieranalysisandthetheoryofpositivedefinitefunctions.
Thismethodmakesitpossibletodescribemuchwiderclassesoffunctionsforwhich
positivityandmonotonicityoccur.Theirresultsincludethetrapezoidalmethodon
acompactintervalandalsoonthewholerealline.
ThechapterbyGuessabandZaimisdevotedtoaunifiedandgeneralframework
forenrichingfiniteelementapproximationsthroughtheuseofadditionalenrichment
functions. They prove a general theorem that characterizes the existence of an
enriched finite element approximation. They also show that their method can
be used to obtain a new class of enriched nonconforming finite elements in
any dimension. For concrete constructions, the authors employ new families of
multivariatetrapezoidal,midpoint,andSimpson-typecubatureformulas.
It is a pleasure to express our gratitude to all the authors and referees without
whose contributions this volume would not have been possible. We would like to
thank Ziqin Feng, Dmitry Glotov, and Eze Nwaze for their help in compiling and
formatting some of the chapters in this volume. Finally, our thanks are due to the
publisherforsupportandcarefulhandlingofthisvolume.
Auburn,AL,USA N.K.Govil
Orlando,FL,USA RamMohapatra
Tuskegee,AL,USA M.A.Qazi
Erlangen,Germany G.Schmeisser