Fuzzy Mathematics: Approximation Theory

446 Pages·2010·5.49 MB·English

by  George A. Anastassiou (auth.)

GeorgeA.Anastassiou
FuzzyMathematics:ApproximationTheory
Studiesin Fuzziness andSoft Computing,Volume251
Editor-in-Chief
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SystemsResearchInstitute
PolishAcademyofSciences
ul.Newelska6
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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
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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