GeorgeA.Anastassiou FuzzyMathematics:ApproximationTheory Studiesin Fuzziness andSoft Computing,Volume251 Editor-in-Chief Prof.JanuszKacprzyk SystemsResearchInstitute PolishAcademyofSciences ul.Newelska6 01-447Warsaw Poland E-mail:[email protected] Furthervolumesofthisseriescanbefoundonourhomepage:springer.com Vol.235.KofiKissiDompere Vol.244.XiaodongLiuandWitoldPedrycz FuzzyRationality,2009 AxiomaticFuzzySetTheoryandIts ISBN978-3-540-88082-0 Applications,2009 ISBN978-3-642-00401-8 Vol.236.KofiKissiDompere EpistemicFoundationsofFuzziness,2009 Vol.245.XuzhuWang,DaRuan, ISBN978-3-540-88084-4 EtienneE.Kerre Vol.237.KofiKissiDompere MathematicsofFuzziness– FuzzinessandApproximateReasoning,2009 BasicIssues,2009 ISBN978-3-540-88086-8 ISBN978-3-540-78310-7 Vol.238.AtanuSengupta,TapanKumarPal Vol.246.PiedadBrox,IluminadaCastillo, FuzzyPreferenceOrderingofInterval SantiagoSánchezSolano NumbersinDecisionProblems,2009 FuzzyLogic-BasedAlgorithmsfor ISBN978-3-540-89914-3 VideoDe-Interlacing,2010 Vol.239.BaodingLiu ISBN978-3-642-10694-1 TheoryandPracticeofUncertain Programming,2009 Vol.247.MichaelGlykas FuzzyCognitiveMaps,2010 ISBN978-3-540-89483-4 ISBN978-3-642-03219-6 Vol.240.AsliCelikyilmaz,I.BurhanTürksen ModelingUncertaintywithFuzzyLogic,2009 Vol.248.Bing-YuanCao ISBN978-3-540-89923-5 OptimalModelsandMethods withFuzzyQuantities,2010 Vol.241.JacekKluska ISBN978-3-642-10710-8 AnalyticalMethodsinFuzzy ModelingandControl,2009 Vol.249.BernadetteBouchon-Meunier, ISBN978-3-540-89926-6 LuisMagdalena,ManuelOjeda-Aciego, Vol.242.YaochuJin,LipoWang José-LuisVerdegay, FuzzySystemsinBioinformatics RonaldR.Yager(Eds.) andComputationalBiology,2009 FoundationsofReasoningunder ISBN978-3-540-89967-9 Uncertainty,2010 ISBN978-3-642-10726-9 Vol.243.RudolfSeising(Ed.) ViewsonFuzzySetsandSystemsfrom Vol.250.XiaoxiaHuang DifferentPerspectives,2009 PortfolioAnalysis,2010 ISBN978-3-540-93801-9 ISBN978-3-642-11213-3 Vol.243.RudolfSeising(Ed.) Vol.251.GeorgeA.Anastassiou ViewsonFuzzySetsandSystemsfrom FuzzyMathematics: DifferentPerspectives,2009 ApproximationTheory,2010 ISBN978-3-540-93801-9 ISBN978-3-642-11219-5 George A. Anastassiou Fuzzy Mathematics: Approximation Theory ABC Author GeorgeA.Anastassiou,Ph.D ProfessorofMathematics TheUniversityofMemphis DepartmentofMathematicalSciences TN38152Memphis USA E-mail:[email protected] ISBN978-3-642-11219-5 e-ISBN978-3-642-11220-1 DOI10.1007/978-3-642-11220-1 StudiesinFuzzinessandSoftComputing ISSN1434-9922 LibraryofCongressControlNumber:2009941640 (cid:2)c 2010Springer-VerlagBerlinHeidelberg Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpart ofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmorinanyother way, and storage in data banks. Duplication of this publication or parts thereof is permittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,in itscurrentversion,andpermissionforuse mustalwaysbeobtainedfrom Springer.ViolationsareliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispub- licationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnames areexemptfromtherelevantprotectivelawsandregulationsandthereforefreefor generaluse. Typeset&CoverDesign:ScientificPublishingServicesPvt.Ltd.,Chennai,India. Printedinacid-freepaper 987654321 springer.com Dedicated to my daughters Angela and Peggy. "No assertion is ever known with certainty ... but that does not stop us making assertions." Carneades, 214-129 BCE "AnyusefullogicmustconcernitselfwithIdeaswithafringeofvagueness and a Truth that is a matter of degree." Norbert Wiener "The facts were always fuzzy or vague or inexact ... Science treated the grayorfuzzyfactsasiftheyweretheblack-whitefactsofmath.Yetnoone had put forth a single fact about the world that was 100% true or 100% false." Bart Kosko, Fuzzy Thinking, 1994, Preface. Preface This monograph is the (cid:133)rst in Fuzzy Approximation Theory. It contains mostly the author(cid:146)s research work on fuzziness of the last ten years and relies a lot on [10]-[32] and it is a natural outgrowth of them. It belongs to the broader area of Fuzzy Mathematics. Chapters are self-contained and several advanced courses can be taught out of this book. WeprovidelotsofapplicationsbutalwayswithintheframeworkofFuzzy Mathematics.Ineachchapterisgivenbackgroundandmotivations. Acom- plete list of references is provided at the end. The topics covered are very diverse. In Chapter 1 we give an extensive basic background on Fuzziness andFuzzyRealAnalysis,aswellacompletedescriptionofthebook.Inthe following Chapters 2,3 we cover in deep Fuzzy Di⁄erentiation and Integra- tion Theory, e.g. we present Fuzzy Taylor Formulae. It follows Chapter 4 on Fuzzy Ostrowski Inequalities. Then in Chapters 5, 6 we present results on classical algebraic and trigonometric polynomial Fuzzy Approximation. In Chapters 7-11 we develop completely the theory of convergence with rates of Fuzzy Positive linear operators to Fuzzy Unit operator, the so called Fuzzy Korovkin Theory. We include there the related topic of Fuzzy Global Smoothness, see Chapter 9. In Chapters 12-14 we deal with Fuzzy WavelettypeoperatorsandtheirconvergencewithratestoFuzzyUnitop- erator. In Chapters 15-16 we discuss similarly as above the Fuzzy Neural Network Operators. In Chapter 17 we deal with Fuzzy Random Korovkin type approximation theory. In Chapter 18 we deal with Fuzzy Random Neural Network approximations. viii Preface In Chapters 19, 20 we present Fuzzy Korovkin type approximations in the Sense of Summability. FinallyinChapter21weestimateinthefuzzysensedi⁄erencesofFuzzy Wavelet type operators. The monograph(cid:146)s approach is Quantitative and almost all main results are given through Fuzzy inequalities, involving fuzzy moduli of continuity, that is fuzzy Jackson type inequalities. Thus all fuzzy convergences are given with rates and the proofs are constructive. The exposed theory is destined and expected to (cid:133)nd applications to all aspects of Fuzziness from theoretical to practical in almost all sciences, technologyandindustry;inourrealworldwemostlyperformfuzzyapprox- imations. On the other hand our theory has its own theoretical merit and interest within the framework of Pure Mathematics. So this monograph is suitable for researchers, graduate students and seminars of theoretical and applied mathematics, computer science, statistics, engineering, etc., also suitable for all science libraries. Fuzzy set theory and applications has experienced a rapid development since its discovery by L. Zadeh in 1965, see [103], its growth and applica- tions now cover almost all kinds of mathematics and applied sciences with great applications to real life. We mention here only a few: (cid:133)nance and stock market, weather prediction, nuclear science, robotics, biomedicine, handwriting analysis, space exploration and satellites, radars, electronics, rheology, agriculture, elevators, ecology, geography and philosophy. For a muchlenghtierlistofapplicationsofFuzzysetsandFuzzylogicseeChapter 1, Section 1.1. IwouldliketothankProfessorSorinGal,UniversityofOradea,Romania, for introducing me into Fuzzy Sets. The (cid:133)nal preparation of book took place during 2009 in Memphis, Ten- nessee,USA.Iwouldliketothankmyfamilyfortheirdedicationandloveto me, which was the strongestsupportduringthe writingof the monograph. I am also indebted and thankful to my graduate student Razvan Mezei for the typing preparation of the manuscript in a short time. November 1, 2009 George A. Anastassiou Department of Mathematical Sciences The University of Memphis, TN, USA Contents 1 INTRODUCTION 1 1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Chapters description . . . . . . . . . . . . . . . . . . . . . . 10 2 ABOUT H-FUZZY DIFFERENTIATION 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 ON FUZZY TAYLOR FORMULAE 51 3.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.2 Addendum . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4 FUZZY OSTROWSKI INEQUALITIES 65 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5 AFUZZYTRIGONOMETRICAPPROXIMATIONTHE- OREM OF WEIERSTRASS-TYPE 75 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 76