Applications of q-Calculus in Operator Theory

274 Pages·2013·1.72 MB·English

by  Ali Aral

Ali Aral · Vijay Gupta
Ravi P. Agarwal
Applications
of q-Calculus
in Operator
Theory
Applications of q-Calculus in Operator Theory

Ali Aral • Vijay Gupta (cid:129) Ravi P. Agarwal
Applications of q-Calculus
in Operator Theory
123
AliAral VijayGupta
DepartmentofMathematics SchoolofAppliedSciences
KırıkkaleUniversity NetajiSubhasInstituteofTechnology
Yahs¸ihan,Kirikkale,Turkey NewDelhi,India
RaviP.Agarwal
DepartmentofMathematics
TexasA&MUniversity-Kingsville
Kingsville,Texas,USA
ISBN978-1-4614-6945-2 ISBN978-1-4614-6946-9(eBook)
DOI10.1007/978-1-4614-6946-9
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013934278
MathematicsSubjectClassification(2010):41A36-41A25-41A17-30E10
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Preface
Simply, quantum calculus is ordinary classical calculus without the notion of
limits. It defines q-calculus and h-calculus. Here h ostensibly stands for Planck’s
constant, while q stands for quantum. A pioneer of q-calculus in approximation
theoryis the formerProfessorAlexandruLupas[117], who firstintroducedtheq-
analogueofBernsteinpolynomials.TenyearslaterPhillips[133]introducedanother
generalization on Bernstein polynomials [113] based on q-integers. Ostrovska
[125,127] studied q-Bernstein polynomials. After that several researchers have
estimatedtheapproximationpropertiesofseveraloperators.Thisbookisanattempt
tocompileandpresentsomepapersonq-calculusinapproximationtheory.
Wedividethebookintosevenchapters.InChap.1,wementionsomenotations
and basic definitions of q-calculus, which will be used throughout the book. We
also present the generating functions of some of the important q-basis functions.
In Chap.2, we present some discrete q-operators, which include the q-Bernstein
polynomials, q-Baskakov operators, q-Sza´sz operators, q-Blemian–Butzer–Hahn
operators,andq-Meyer–Ko¨nigandZelleroperators.Wepresenttheapproximation
propertiesofsuchoperators.
In Chap.3, we present the q-analogue of integral operators which include q-
Picard and q-Weierstrass-type singular integral operators and study their rate of
convergenceandweightapproximation.Wealsodiscusserrorestimationandglobal
smoothness preservation property of such operators. In the last section of this
chapter,westudygeneralizedPicardoperatorsandpointwiseconvergence,orderof
pointwiseconvergence,andnormconvergenceofthegeneralizedoperators.Inthe
lastsection,westudytheq-Meyer–Ko¨nig–Zeller–Durrmeyeroperatorsandestimate
themomentsandsomedirectresults.
In Chap.4, we study the integral modifications of Bernstein operators using
the q-beta functions of the first kind. We present the approximation properties of
the q-Bernstein–Kantorovichoperators, q-Bernstein–Durrmeyerpolynomials, dis-
cretely defined q-Durrmeyer-type operators, and genuine q-Bernstein–Durrmeyer
operators.Wementionthemomentestimation,directresults,andthelimitingcon-
vergenceofsuchoperators.Wehavealsoincludedasectiononfuzzyapproximation
andapplications.
v
vi Preface
InChap.5,wediscusssomeotherrecentlyintroducedq-integraloperatorsonthe
positive real axis. To tackle such operators, we generally use q-beta functions of
thesecondkind.Thischapterincludesq-Baskakov–Durrmeyeroperators,q-Sza´sz-
betaoperators,q-Sza´sz–Durrmeyeroperators,andq-Phillipsoperators.Wepresent
moments, recurrence relations for moments, asymptotic formula, and weighted
approximationsforsuchoperators.
InChap.6,westudythestatisticalconvergenceoftheq-operators.Wemention
results for a general class of positive linear operators and present statistical
approximationpropertiesinweightedspace.Wealsopresenttheresultsforq-Sza´sz–
King-type operatorsand q-Baskakov–Kantorovichoperatorsand the study rate of
convergence.
In the last chapter, we present the quantitative Voronovskaja-type estimate for
certainq-Durrmeyerpolynomials.Inthis way,we putin evidencethe overconver-
gencephenomenonforthese q-Durrmeyerpolynomials,namely,the extensionsof
approximationproperties(with quantitative estimates) from the real interval [0,1]
to compact disks in the complex plane. Also, we study the complex q-Gauss–
Weierstrass integraloperators.We showthattheseoperatorsare anapproximation
process in some subclasses of analytic functions giving Jackson-type estimates
in approximation. Furthermore, we give q-calculus analogues of some shape-
preservingpropertiesfortheseoperatorssatisfiedbytheclassicalcomplexGauss–
Weierstrassintegraloperators.
Kirikkale,Turkey AliAral
NewDelhi,India VijayGupta
Kingsville,TX RaviP.Agarwal
Contents
Introduction ...................................................................... xi
1 Introductionofq-Calculus.................................................. 1
1.1 NotationsandDefinitionsinq-Calculus............................... 1
1.2 q-Derivative ............................................................. 3
1.3 q-SeriesExpansions..................................................... 5
1.4 GeneratingFunctions................................................... 8
1.4.1 GeneratingFunctionforq-BernsteinBasis.................... 9
1.4.2 GeneratingFunctionforq-MKZ............................... 10
1.4.3 GeneratingFunctionforq-BetaBasis ......................... 10
1.5 q-Integral................................................................ 11
2 q-DiscreteOperatorsandTheirResults................................... 15
2.1 q-BernsteinOperators................................................... 15
2.1.1 Introduction..................................................... 16
2.1.2 BernsteinPolynomials.......................................... 16
2.1.3 Convergence .................................................... 18
2.1.4 Voronovskaya’sTheorem....................................... 20
2.2 q-Sza´szOperators....................................................... 22
2.2.1 Introduction..................................................... 23
2.2.2 ConstructionofOperators...................................... 23
2.2.3 AuxiliaryResult ................................................ 24
2.2.4 ConvergenceofSqn(f) ......................................... 25
n
2.2.5 ConvergencePropertiesinWeightedSpace................... 29
2.2.6 OtherProperties................................................. 32
2.3 q-BaskakovOperators .................................................. 35
2.3.1 ConstructionofOperatorsandSomePropertiesofThem.... 35
2.3.2 ApproximationProperties...................................... 40
2.3.3 Shape-PreservingProperties ................................... 42
2.3.4 MonotonicityProperty ......................................... 45
vii
viii Contents
2.4 ApproximationPropertiesofq-BaskakovOperators.................. 48
2.4.1 Introduction..................................................... 48
2.4.2 MainResults.................................................... 50
2.4.3 Proofs............................................................ 50
2.5 q-Bleimann–Butzer–HahnOperators .................................. 62
2.5.1 Introduction..................................................... 63
2.5.2 ConstructionoftheOperators.................................. 63
2.5.3 PropertiesoftheOperators..................................... 65
2.5.4 SomeGeneralizationofL ..................................... 71
n
3 q-IntegralOperators......................................................... 73
3.1 q-Picardandq-Gauss–WeierstrassSingularIntegralOperator ....... 73
3.1.1 Introduction..................................................... 73
3.1.2 RateofConvergenceinL (R)................................. 75
p
3.1.3 ConvergenceinWeightedSpace............................... 78
3.1.4 ApproximationError ........................................... 80
3.1.5 GlobalSmoothnessPreservationProperty .................... 84
3.2 GeneralizedPicardOperators .......................................... 85
3.2.1 Introduction..................................................... 86
3.2.2 PointwiseConvergence......................................... 91
3.2.3 OrderofPointwiseConvergence............................... 95
3.2.4 NormConvergence............................................. 98
3.3 q-Meyer–Ko¨nig–Zeller–DurrmeyerOperators ........................ 100
3.3.1 Introduction..................................................... 101
3.3.2 EstimationofMoments......................................... 101
3.3.3 Convergence .................................................... 106
4 q-Bernstein-TypeIntegralOperators...................................... 113
4.1 Introduction ............................................................. 113
4.2 q-Bernstein–KantorovichOperators ................................... 113
4.2.1 DirectResults................................................... 114
4.3 q-Bernstein–DurrmeyerOperators..................................... 117
4.3.1 AuxiliaryResults ............................................... 117
4.3.2 DirectResults................................................... 121
4.3.3 ApplicationstoRandomandFuzzyApproximation.......... 127
4.4 DiscretelyDefinedq-DurrmeyerOperators............................ 132
4.4.1 MomentEstimation............................................. 132
4.4.2 RateofApproximation......................................... 134
4.5 Genuineq-Bernstein–DurrmeyerOperators........................... 139
4.5.1 Moments ........................................................ 139
4.5.2 DirectResults................................................... 140
Contents ix
4.6 q-BernsteinJacobiOperators........................................... 141
4.6.1 BasicResults.................................................... 142
4.6.2 Convergence .................................................... 143
5 q-Summation–IntegralOperators.......................................... 145
5.1 q-Baskakov–DurrmeyerOperators..................................... 145
5.1.1 ConstructionofOperators...................................... 146
5.1.2 LocalApproximation........................................... 151
5.1.3 RateofConvergence............................................ 154
5.1.4 WeightedApproximation....................................... 155
5.1.5 RecurrenceRelationandAsymptoticFormula................ 157
5.2 q-Sza´sz-BetaOperators................................................. 164
5.2.1 ConstructionofOperators...................................... 164
5.2.2 DirectTheorem................................................. 167
5.2.3 WeightedApproximation....................................... 170
5.3 q-Sza´sz–DurrmeyerOperators ......................................... 171
5.3.1 AuxiliaryResults ............................................... 172
5.3.2 ApproximationProperties...................................... 175
5.4 q-PhillipsOperators..................................................... 180
5.4.1 Moments ........................................................ 181
5.4.2 DirectResults................................................... 184
5.4.3 Voronovskaja-TypeTheorem................................... 190
6 StatisticalConvergenceofq-Operators.................................... 195
6.1 GeneralClassofPositiveLinearOperators............................ 196
6.1.1 NotationsandPreliminaryResults............................. 196
6.1.2 ConstructionoftheOperators.................................. 196
6.1.3 StatisticalApproximationPropertiesinWeightedSpace..... 200
6.1.4 SpecialCasesofT Operator................................... 203
n
6.2 q-Sza´sz–King-typeOperators .......................................... 205
6.2.1 NotationsandPreliminaries.................................... 205
6.2.2 WeightedStatisticalApproximationProperty................. 207
6.2.3 RateofWeightedApproximation.............................. 209
6.3 q-Baskakov–KantorovichOperators................................... 213
6.3.1 Introduction..................................................... 213
6.3.2 q-AnalogueofBaskakov–KantorovichOperators ............ 215
6.3.3 WeightedStatisticalApproximationProperties............... 217
6.3.4 RateofConvergence............................................ 218
7 q-ComplexOperators........................................................ 223
7.1 Summation-Integral-TypeOperatorsinCompactDisks .............. 223
7.1.1 BasicResults.................................................... 224
7.1.2 UpperBound.................................................... 228
7.1.3 AsymptoticFormulaandExactOrder......................... 230
7.2 q-Gauss–WeierstrassOperator ......................................... 238
7.2.1 Introduction..................................................... 238