Fuzzy Mathematics: Approximation Theory

This monograph is the r st in Fuzzy Approximation Theory. It contains mostly the author 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. We provide lots of applications but always within the framework of Fuzzy Mathematics. In each chapter is given background and motivations. A c- 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 and Fuzzy Real Analysis, as well a complete description of the book. In the following Chapters 2,3 we cover in deep Fuzzy Di?erentiation and Integ- 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.

The theory presented in this book is destined and expected to find applications to all aspects of fuzziness from theoretical to practical in almost all sciences, technology, finance and industry, as well as within pure mathematics.

This monograph is the r st in Fuzzy Approximation Theory. It contains mostly the author 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. We provide lots of applications but always within the framework of Fuzzy Mathematics. In each chapter is given background and motivations. A c- 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 and Fuzzy Real Analysis, as well a complete description of the book. In the following Chapters 2,3 we cover in deep Fuzzy Di?erentiation and Integ- 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.

ABOUT H-FUZZY DIFFERENTIATION.- ON FUZZY TAYLOR FORMULAE.- FUZZY OSTROWSKI INEQUALITIES.- A FUZZY TRIGONOMETRIC APPROXIMATION THEOREM OF WEIERSTRASS-TYPE.- ON BEST APPROXIMATION AND JACKSON-TYPE ESTIMATES BY GENERALIZED FUZZY POLYNOMIALS.- BASIC FUZZY KOROVKIN THEORY.- FUZZY TRIGONOMETRIC KOROVKIN THEORY.- FUZZY GLOBAL SMOOTHNESS PRESERVATION.- FUZZY KOROVKIN THEORY AND INEQUALITIES.- HIGHER ORDER FUZZY KOROVKIN THEORY USING INEQUALITIES.- FUZZY WAVELET LIKE OPERATORS.- ESTIMATES TO DISTANCES BETWEEN FUZZY WAVELET LIKE OPERATORS.- FUZZY APPROXIMATION BY FUZZY CONVOLUTION OPERATORS.- DEGREE OF APPROXIMATION OF FUZZY NEURAL NETWORK OPERATORS, UNIVARIATE CASE.- HIGHER DEGREE OF FUZZY APPROXIMATION BY FUZZY WAVELET TYPE AND NEURAL NETWORK OPERATORS.- FUZZY RANDOM KOROVKIN THEOREMS AND INEQUALITIES.- FUZZY-RANDOM NEURAL NETWORK APPROXIMATION OPERATORS, UNIVARIATE CASE.- -SUMMABILITY AND FUZZY KOROVKIN APPROXIMATION.- -SUMMABILITY AND FUZZY TRIGONOMETRIC KOROVKIN APPROXIMATION.- UNIFORM REAL AND FUZZY ESTIMATES FOR DISTANCES BETWEEN WAVELET TYPE OPERATORS AT REAL AND FUZZY ENVIRONMENT.