Approximation and computation: In honor of Gradimir V. Milovanovic

498 Pages·2011·3.62 MB·English

by  Aleksandar Ivić (auth.)

APPROXIMATION AND COMPUTATION
Formoretitlesinthisseries,goto
http://www.springer.com/series/7393
Springer Optimization and Its Applications
VOLUME42
ManagingEditor
PanosM.Pardalos(UniversityofFlorida)
Editor—CombinatorialOptimization
Ding-ZhuDu(UniversityofTexasatDallas)
AdvisoryBoard
J.Birge(UniversityofChicago)
C.A.Floudas(PrincetonUniversity)
F.Giannessi(UniversityofPisa)
H.D.Sherali(VirginiaPolytechnicandStateUniversity)
T.Terlaky(McMasterUniversity)
Y.Ye(StanfordUniversity)
AimsandScope
Optimizationhasbeenexpandinginalldirectionsatanastonishingratedur-
ing the last few decades. New algorithmic and theoretical techniques have
beendeveloped,thediffusionintootherdisciplineshasproceededatarapid
pace, and our knowledge of all aspects of the field has grown even more
profound.At thesametime, oneofthe moststriking trendsin optimization
is the constantly increasing emphasis on the interdisciplinary nature of the
field.Optimizationhasbeenabasictoolinallareasofappliedmathematics,
engineering,medicine,economicsandothersciences.
The series Springer Optimization and Its Applications publishes under-
graduate and graduate textbooks, monographs and state-of-the-art exposi-
tory works that focus on algorithms for solving optimization problems and
alsostudyapplicationsinvolvingsuchproblems.Someofthetopicscovered
includenonlinearoptimization(convexandnonconvex),networkflowprob-
lems, stochastic optimization, optimal control, discrete optimization, mul-
tiobjective programming, description of software packages, approximation
techniquesandheuristicapproaches.
APPROXIMATION AND COMPUTATION
In Honor of Gradimir V. Milovanovic´
EditedBy
WALTERGAUTSCHI
PurdueUniversity
DepartmentofComputerScience
WestLafayette,Indiana,USA
GIUSEPPEMASTROIANNI
UniversityofBasilicata
DepartmentofMathematicsandComputerSciences
Potenza,Italy
THEMISTOCLESM.RASSIAS
NationalTechnicalUniversityofAthens
DepartmentofMathematics
Athens,Greece
ABC
Editors
WalterGautschi ThemistoclesM.Rassias
PurdueUniversity NationalTechnicalUniversityofAthens
DepartmentofComputerScience DepartmentofMathematics
305N.UniversityStreet ZografouCampus
WestLafayette,Indiana,47907 15780Athens
USA Greece
[email protected] [email protected]
GiuseppeMastroianni
UniversityofBasilicata
DepartmentofMathematicsand
ComputerSciences
Vialedell’AteneoLucano,10
85100Potenza
Italy
[email protected]
ISSN1931-6828
ISBN978-1-4419-6593-6 e-ISBN978-1-4419-6594-3
DOI10.1007/978-1-4419-6594-3
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2010937561
MathematicsSubjectClassification(2010):37D40,65Kxx,68U35,68Txx
(cid:2)c SpringerScience+BusinessMedia,LLC2011
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Preface
The book “APPROXIMATION AND COMPUTATION” deals with recent results
in approximationtheory, numericalanalysis, and various applications of an inter-
disciplinary character. It contains papers written by experts from 15 countries in
theirrespectivesubjects.Mostofthese paperswere presentedin personduringan
InternationalConference,whichwasheldattheUniversityofNisˇ, Serbia(August
25–29,2008),tohonorthe60thanniversaryofthewell-knownmathematicianPro-
fessor Gradimir V. Milovanovic´.Professor Milovanovic´,nextto other distinctions
he has received, is a correspondingmember of the Serbian Academy of Sciences
andArts.
ThebookconsistsofthefollowingfiveParts:
1. Introduction
2. PolynomialsandOrthogonalSystems
3. QuadratureFormulae
4. DifferentialEquations
5. Applications
Ineachpart,thepapersappearinalphabeticalorderwithrespecttothelastname
of the first-named author, except in Part 1. The latter contains three contributions
connectedwiththescientificworkofG.V.Milovanovic´.Besidesbiographicaldata,
A.Ivic´presentscontributionsofMilovanovic´inthefieldofpolynomialsandquadra-
tureprocesses.W.GautschidescribeshiscollaborationwithMilovanovic´overmany
years,andTh.M.Rassiasgivesanaccountonsomemajortrendsinmathematics.
Polynomials,algebraicandtrigonometric,andcorrespondingorthogonalsystems
are treated as basic constructive elements in Part 2. P. Barry, P.M. Rajkovic´, and
M.D.Petkovic´ giveanapplicationofSobolevorthogonalpolynomialstothecom-
putation of a special Hankel determinant. Extremal problems for polynomials in
the complex plane, including the well-known conjecture of Sendov about critical
points of algebraic polynomials and the mean value conjecture of Smale, as well
as certain relations between these two famous open problems, are considered by
B. Bojanov. He also formulates a conjecture that seems to be a natural complex
v
vi Preface
analog of Rolle’s theorem and contains as a particular case Smale’s conjecture.
V. Bozˇin and M. Mateljevic´ characterize graphs of maximal energy by means of
orthogonalmatrices.Theirresultmakesitpossibletoestimatetheenergyofgraphs
without direct computationof eigenvalues. A.S. Cvetkovic´ further develops inter-
lacingpropertiesofzerosofshiftedJacobipolynomials,theinvestigationofwhich
hasrecentlybeeninitiatedbyK.DriverandK.Jordaan.He,infact,provescertain
improvementsoftheirresults.A.S.Cvetkovic´ andM.P.Stanic´ investigatetrigono-
metricpolynomialsofsemi-integerdegreewithrespecttosomeweightfunctionson
[−π,π)andorthogonalsystemsconnectedwithinterpolatoryquadratureruleswith
anevenmaximaltrigonometricdegreeofexactness.W. Gautschigivesanaccount
of computational work in support of conjectured inequalities for zeros of Jacobi
polynomials, the sharpness of Bernstein’s inequality for Jacobi polynomials, and
thepositivityofcertainquadratureformulaeofNewton–Cotes,Gauss–Radau,and
Gauss–Lobattotype.Theuseofsymboliccomputationisdescribedforgenerating
Gaussquadratureruleswithexoticweightfunctions,specifically weightfunctions
decayingsuper-exponentiallyatinfinity,andweightfunctionsdenselyoscillatoryat
zero.J.GilewiczandR.Jedynakilluminatethecompatibilityofcontinuedfraction
convergentswithPade´approximants.Orthogonaldecompositionsoffractalsetsare
considered by Lj.M. Kocic´, S. Gegovska-Zajkova,and E. Babacˇe. The last paper
inthispartbyS.Koumandosgivesasystematicaccountofnewresultsonpositive
trigonometricsums and applicationsto geometricfunctiontheory.His work is re-
latedtorecentinvestigationsconcerningsharpeningandgeneralizationsofthecele-
bratedVietorisinequalities.Far-reachingextensionsandresultsonstarlikefunctions
areobtainedandnewpositivesumsofGegenbauerpolynomialsandadiscussionof
somechallengingconjecturesarepresented.
Part3isdedicatedtonumericalintegration,includingquadratureformulaswith
equidistant nodes and formulas of Gaussian type having multiple nodes. Several
quadraturerulesforthenumericalintegrationofsmooth(nonoscillatory)functions,
definedonthereal(positive)semiaxisorontherealaxisanddecayingalgebraically
atinfinity,areexaminedbyG.MonegatoandL.Scuderi.Amongthoseconsidered
on the real axis, there are four new alternative numericalapproaches. The advan-
tages and the disadvantages of each of them are pointed out through several nu-
mericaltests,involvingeitherthecomputationofasingleintegralorthenumerical
solution of some integral equations. G. Nikolov and C. Simian construct Gauss,
Lobatto, and Radau quadrature formulae associated with the spaces of parabolic
splineswithequidistantknots.Thesequadratureformulaeareknowntobeasymp-
totically optimal in Sobolev spaces W3. Sharp estimates for the error constant in
p
W∞3 are given.Somequadraturerulesbasedonthe zerosof Freudpolynomialsfor
computingCauchyprincipalvalueintegralsonthereallineareproposedbyI.No-
tarangelo. M.M. Spalevic´, and M.S. Pranic´ give a survey on contour integration
methods for estimating the remainder of Gauss–Tura´n quadrature rules involving
analyticfunctions.Finally,J.Waldvogelproposestousethetrapezoidalruleonthe
entire real line R as the standard algorithm for numerical quadrature of analytic
functions.Otherintervalsandslowlydecayingintegrandsmayelegantlybehandled
bymeansofsimpleanalytictransformationsoftheintegrationvariable.
Preface vii
Methods for differential equations are considered in Part 4. D.R. Bojovic´ and
B.S. Jovanovic´ investigate the convergence of difference schemes for the one-
dimensionalheatequationwith time-dependentoperatorandthe coefficientofthe
time derivativecontaining a Dirac delta distribution. An abstract operator method
isdevelopedforanalyzingthisequation.InthepaperbyS.Burke,Ch.Ortner,and
E. Su¨li, the energy of the Francfort–Marigo model of brittle fracture is approxi-
mated,inthesenseofΓ-convergence,bytheAmbrosio–Tortorellifunctional.They
formulateand analyze an adaptivefinite elementalgorithm,combiningan inexact
Newton method with residual-driven adaptive mesh refinement, for the computa-
tionofits(local)minimizers.Thesequencegeneratedbythisalgorithmisprovedto
convergetoa criticalpoint.C. Frammartinoproposesa Nystro¨mmethodforsolv-
ing integralequationsequivalentto second-orderboundaryvalue problemson the
real semiaxis. She proves the stability and convergence of such a procedure and
gives some interesting numerical examples. B.S. Jovanovic´ investigates an initial
boundaryvalueproblemforaone-dimensionalhyperbolicequationintwodisjoint
intervals.Afinitedifferenceschemeapproximatingthisinitialboundaryvalueprob-
lemisproposedandanalyzed.P.S.Milojevic´ developsanonlinearFredholmalter-
nativetheoryinvolvingk-ballandk-setperturbationsof generalhomeomorphisms
as well as of homeomorphisms that are nonlinear Fredholm maps of index zero.
He gives several applications to the unique and finite solvability of potential and
semilinearproblemswithstronglynonlinearboundaryconditionsandtoquasilinear
elliptic equations. The work of S. Pilipovic´, N. Teofanov, and J. Toft shows pos-
sibledirectionsfornumericallyinterestedmathematicianstoapproximatedifferent
typesofsingularsupports,wavefrontsets, andpseudodifferentialoperatorsinthe
frameworkofFourier-Lebesguespaces.Itcontainsnewresultsonsingularsupports
in Fourier-Lebesguespaces and on the continuity propertiesof certain pseudodif-
ferentialoperators.Finally,B.M.Piperevskiconsidersaclassofmatrixdifferential
equationsandgivesconditionsunderwhichthisclasshasapolynomialsolution.
Finally,contributionsdiscussingvariousapplicationsarepresentedinPart5.An
improvedalgorithmforPetviashvili’s heuristic numericalmethodfor findingsoli-
tonsinopticallyinducedphotoniclatticesispresentedbyR.Jovanovic´andM.Tuba.
An explicit method for the numerical solution of the Fokker–Planck equation of
filtered phase noise in modern telecommunication systems is given by D. Milic´.
Z.S.Nikolic´investigatesnumericallythedensificationduetogravity-inducedskele-
talsettlingduringliquidphasesintering.Anewmethodologyisappliedforsimula-
tionofmicrostructuralevolutionofa regularmultidomainmodel.P.Stanimirovic´,
M. Miladinovic´,and I.M. Jovanovic´ investigatesymbolic transformationson “un-
evaluated” expressions representing objective functions to generate unevaluated
compositeobjectivefunctionsrequiredduringtheimplementationofunconstrained
nonlinearoptimizationmethodsbased onthe exactline search. N. Stevanovic´ and
P.V.Protic´ dealwithAbel-Grassmann’sgroupoidsandintroducetheclass“rootof
aband”,ageneralizationofAG-bandandAG-3-band.B.T.Todorovic´,S.R.Rancˇic´,
andE.H. Mulalic´ consideraversionofprobabilisticsupervisedmachine,learning
classifierinnamedentityrecognition.Z.Udovicˇic´dealswithinterpolatingquadratic
viii Preface
splines,andfinally,Lj.S. Velimirovic´,S.R. Rancˇic´,andM.Lj. Zlatanovic´consider
infinitesimalbendingofcurvesinE3.
Thisbookaddressesresearchersandstudentsinmathematics,physics,andother
computationalandappliedsciences.
AscoauthorsofProfessorGradimirV.Milovanovic´,wefeelverypleasedtohave
undertakenthepreparationofthispublication.
Finally, we expressour warmest thanksto all of the scientists who contributed
to this volume, to the referees for their carefulreading of the manuscripts, and to
collaboratorsofProfessorMilovanovic´:LjubisˇaKocic´,AleksandarCvetkovic´,and
MarijaStanic´,whohelpedintheorganizationoftheConferenceandintheprepara-
tionofthisvolume.
WestLafayette/Potenza/Athens WalterGautschi
August,2009 GiuseppeV.Mastroianni
ThemistoclesM.Rassias
Contents
Preface............................................................ v
PartI Introduction
TheScientificWorkofGradimirV.Milovanovic´ ..................... 3
AleksandarIvic´
1 Introduction.............................................. 3
2 TheBiographyofG.V.Milovanovic´.......................... 3
3 FieldsofScientificWorkofGVM ........................... 5
4 GVMandQuadratureProcesses............................. 6
4.1 ConstructionofGaussianQuadratures ................ 6
4.2 Moment-PreservingSpline Approximationand
Quadratures ...................................... 8
4.3 QuadratureswithMultipleNodes .................... 9
4.4 OrthogonalitywithRespecttoaMomentFunctional
andCorrespondingQuadratures ..................... 10
4.5 NonstandardQuadraturesofGaussianType............ 13
4.6 GaussianQuadratureforMu¨ntzSystems .............. 15
5 GVMandPolynomials..................................... 16
5.1 ClassicalOrthogonalPolynomials.................... 16
5.2 ExtremalProblemsofMarkov–BernsteinTypefor
Polynomials...................................... 17
5.3 OrthogonalPolynomialsonRadialRays .............. 21
6 GVMandInterpolationProcesses ........................... 24
7 TheSelectedBibliographyofGVM.......................... 25
References..................................................... 25
MyCollaborationwithGradimirV.Milovanovic´ ..................... 33
WalterGautschi
References..................................................... 42
ix