Convergence Estimates in Approximation Theory

368 Pages·2014·2.88 MB·English

by  Vijay Gupta

Vijay Gupta · Ravi P. Agarwal
Convergence
Estimates in
Approximation
Theory
Convergence Estimates in Approximation Theory

Vijay Gupta • Ravi P. Agarwal
Convergence Estimates
in Approximation Theory
123
VijayGupta RaviP.Agarwal
SchoolofAppliedSciences DepartmentofMathematics
NetajiSubhasInstituteofTechnology TexasA&MUniversity-Kingsville
NewDelhi,India Kingsville,TX,USA
ISBN978-3-319-02764-7 ISBN978-3-319-02765-4(eBook)
DOI10.1007/978-3-319-02765-4
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013956134
MathematicsSubjectClassification(2010):41A36,41A25,41A17,30E10
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Preface
Theaimofthegeneralapproximationmethodsconcerninglinearpositiveoperators
istodealwithconvergencebehavior.Theaccuracycanbeascertainedtoadesired
degree by applying different methods. We are also concerned with the amountof
computationrequiredtoachievethisaccuracy.Adirecttheoremprovidestheorder
ofapproximationforfunctionsofspecifiedsmoothness.Theconverseofthedirect
result,thatis, the inversetheorem,infersthe natureofsmoothnessofthe function
fromitsorderofapproximation.Asymptoticanalysisisakeytoolforexploringthe
ordinaryandpartialdifferentialequationsthatarise in the mathematicalmodeling
of real-world phenomena. The rate of convergence infers the speed at which a
convergentsequenceapproachesitslimit.
This work treats the convergence results mainly for linear positive operators.
After the well-known theorem due to Weierstrass and the important convergence
theoremofKorovkin,manynewoperatorswereproposedandconstructedbyseveral
researchers.Thetheoryoftheseoperatorshasbeenanimportantareaofresearchin
thelastfewdecades.Thebasicresultsanddirectestimatesinbothlocalandglobal
approximationarealsopresented.Wealsodiscusstheasymptoticexpansionofsome
of the linear positive operators,which is importantfor convergenceestimates. We
knowthattoimprovetheorderofapproximation,wecanconsiderthecombinations.
Byconsideringthelinearcombinations,wehavetoslackenthepositivityconditions
of the operators. Here we discuss linear and iterative combinations and present
someresults.Someoperatorsreproduceconstantandlinearfunctions.Twodecades
agoitwas observedthatif we modifythe originaloperators,we canhave a better
approximation.Latersomeresearchersstudiedthisdirectionandobservedthatsome
operatorsthatdonotevenpreservelinearfunctionscangiveabetterapproximation
iftheyaremodifiedtopreservelinearfunctions.Theoverconvergencephenomena
for certain operators, which were not discussed in the book by Gal, [77] are also
discussed here. We presentsome results of overconvergenciesto largersets in the
complexplane.
Inthisbook,thecrucialroleoftherateofconvergenceforfunctionsofbounded
variationandforfunctionshavingderivativesofboundedvariationisemphasized.
New and efficient methods that are applicable to general operators are also
v
vi Preface
discussed.Theadvantagesofthesemethodsconsistofobtainingimprovedandeven
optimalestimates,aswellasofbroadeningtheapplicabilityoftheresults.Several
results have been established for different exponential-typeoperators and integral
operators. Still, this type of study can be extended to other generalizationsof the
knownoperators.Also,therateofconvergenceforfunctionsofboundedvariation
on mixed summation–integral-typeoperators having different basis functions has
not been studied to date, including Szász–Baskakov, Baskakov–Szász, and Beta–
Szászoperators,amongothers.Itcanbeconsideredanopenproblem.Wemention
someresultswithoutproofs,whileinothercases,proofsandoutlinesaregiven.The
bookisusefulforbeginnersandforthosewhoareworkingonanalysisandrelated
areas.
Thismonographisdividedinto11chapters:
InChap.1,whichisintroductoryinnature,wepresentsomebasicdefinitionsand
thestandardtheorems,whichareimportantfortheconvergencepointofview.
In Chap.2, we discuss some important results on linear positive operators
relatedtoconvergence.Somedirectresults,whichincludepointwiseconvergence,
asymptotic formulas, and error estimations, are presented here. We also deal
with some important results, including discretely defined operators, Kantorovich-
and Durrmeyer-typeoperators,mixed summation–integral-typeoperators,Phillips
operators,andotherintegral-typeoperators.
In Chap.3, we discuss the asymptotic behavior of some of the linear positive
operators. We discuss the complete asymptotic expansion of the Baskakov–
Kantorovich, Szász–Baskakov, Meyer–König–Zeller–Durrmeyer, and Beta
operators.
In Chap.4, we mention approximation for certain combinations. The linear
combinations are no longer positive operators. Here we study some results for
linear and iterative combinations.Also, we consider a differentform of the linear
combinations and present direct estimates for combinations of Szász–Baskakov
operators.
In Chap.5,we presentthe techniquesforgettinga better approximation.Many
well-known approximating operators L ; preserve the constant as well as linear
n
functions. In 1983, King considered the modification of the classical Bernstein
polynomials so that the modified form preserves the test function e : A better
2
approximationcan be achieved for the modified form. In this chapter, we present
someresultsfordifferentoperatorsdiscussedinrecentyears.
In Chap.6, we study the overconvergence phenomenon for certain operators
that were not discussed in the book by Gal [77]. We present some results of
overconvergenciestolargersetsin the complexplanethanin the realdomain.We
discusssomeveryrecentresultsoncomplexBaskakov–Stancuoperators,complex
Favard–Szász–Mirakjan–Stancu operators, complex Beta operators of the second
kind, genuine Durrmeyer–Stancu polynomials, and certain complex Durrmeyer-
typeoperators.
In Chap.7, the pointwise approximation properties of some approximation
operators of the probabilistic type are established and studied. The rates of
convergenceforLegendre–Fourierseries,Hermite–Fejérpolynomials,exponential-
Preface vii
typeoperators(whichincludeBernstein,Szász,Baskakov,Gamma,andWeierstrass
operators) are also discussed. The rates of convergence of such operators for a
bounded variation function f are given at those points x; where f.xC/ and
f.x(cid:2)/exist.HerewealsopresentsomeresultsontheDurrmeyervariantsofsuch
operators. We also present the related results for a general class of summation–
integral-typeoperatorsandtheMeyer–KönigandZelleroperators.
In Chap.8, we discuss various Bézier variants of the approximation opera-
tors. They have close relationships with the fields of geometry modeling and
design. In this chapter, we present some important results on the approximation
of Bézier variants of a series of approximation operators, which include Bézier
variants of the Bernstein, Bleimann–Butzer–Hann, Balazs–Kantorovich, Szász–
Kantorovich,Baskakov, Baskakov–Kantorovich,Baskakov–Durrmeyer,and MKZ
operators,amongothers.
In Chap.9, we study some otherrelated results. We presentthe rate of approx-
imation on nonlinearoperators.Also, we discuss the rate of convergencein terms
of Chanturiya’s modulus of variation. We also present some results for bounded
andabsolutelycontinuousfunctions.Finally,wepresenttherateofconvergencefor
functionshavingderivativescoincidinga.e.withthefunctionofboundedvariation.
InChap.10,wepresentresultsrelatedtosimultaneousapproximation.Although
lot of papers have appeared on such topic, in this chapter we mention the rate of
simultaneousapproximationfor certain Durrmeyer–Bézieroperatorsfor functions
of bounded variation. We also present the rate of convergence in simultaneous
approximationforcertainsummation–integral-typeoperatorshavingderivativesof
boundedvariation.
Inthebrieffinalchapter,Chap.11,wepresentthefuturescopeofrelatedtopics
andmentionsomeopenproblems.
NewDelhi,India VijayGupta
Kingsville,TX,USA RaviP.Agarwal

Acknowledgments
Idedicatethisbookinthememoryofmyfather,thelateP.D.Gupta(1935–1996),
whoinspiredandencouragedmeto optformathematicsasmymainsubjectatan
earlystage.I amindebtedto mymother,UmaGupta,forherconstantblessings.I
appreciate the constant support of my wife, Shalini, and my children, Arushi and
Anshay,ateachstageofthisbook’swriting.
-VijayGupta
Idedicatethisbookinthememoryofmyfather,thelateRadheyShiamAgarwal.
-RaviP.Agarwal
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