This monograph belongs to the broader area of Fuzzy Mathematics and it is the first one in Fuzzy Approximation Theory. The chapters are self-contained with lots of applications to teach several advanced courses and the topics covered are very diverse. An extensive background of Fuzziness and Fuzzy Real Analysis is given. The author covers Fuzzy Differentiation and Integration Theory followed by Fuzzy Ostrowski inequalities. Then results on classical algebraic and trigonometric polynomial Fuzzy Approximation are presented. The author develops a complete theory of convergence with rates of Fuzzy Positive linear operators to Fuzzy unit operator, the so-called Fuzzy Korovkin Theory. The related Fuzzy Global Smoothness is included. Then follows the study of Fuzzy Wavelet type operators and their convergence with rates to Fuzzy unit operator. Similarly the Fuzzy Neural Network Operators are discussed followed by Fuzzy Random Korovkin approximation theory and Fuzzy Random Neural Network approximations. The author continues with Fuzzy Korovkin approximations in the sense of Summability. Finally fuzzy sense differences of Fuzzy Wavelet type operators are estimated.
The monograph's approach is quantitative and the main results are given via Fuzzy inequalities, involving Fuzzy moduli of continuity, that is Fuzzy Jackson type inequalities.
The exposed theory is destined and expected to find applications to all aspects of Fuzziness from theoretical to practical in almost all sciences, technology, finance and industry. Also it has its interest within Pure Mathematics. So this monograph is suitable for researchers, graduate students and seminars of theoretical and applied mathematics, computer science, statistics and engineering.
Author(s): George A. Anastassiou (auth.)
Series: Studies in Fuzziness and Soft Computing 251
Edition: 1
Publisher: Springer-Verlag Berlin Heidelberg
Year: 2010
Language: English
Pages: 444
Tags: Computational Intelligence; Artificial Intelligence (incl. Robotics); Probability Theory and Stochastic Processes
Front Matter....Pages -
INTRODUCTION....Pages 1-14
ABOUT H-FUZZY DIFFERENTIATION....Pages 15-50
ON FUZZY TAYLOR FORMULAE....Pages 51-63
FUZZY OSTROWSKI INEQUALITIES....Pages 65-73
A FUZZY TRIGONOMETRIC APPROXIMATION THEOREM OF WEIERSTRASS-TYPE....Pages 75-82
ON BEST APPROXIMATION AND JACKSON-TYPE ESTIMATES BY GENERALIZED FUZZY POLYNOMIALS....Pages 83-97
BASIC FUZZY KOROVKIN THEORY....Pages 99-104
FUZZY TRIGONOMETRIC KOROVKIN THEORY....Pages 105-113
FUZZY GLOBAL SMOOTHNESS PRESERVATION....Pages 115-124
FUZZY KOROVKIN THEORY AND INEQUALITIES....Pages 125-157
HIGHER ORDER FUZZY KOROVKIN THEORY USING INEQUALITIES....Pages 159-190
FUZZY WAVELET LIKE OPERATORS....Pages 191-207
ESTIMATES TO DISTANCES BETWEEN FUZZY WAVELET LIKE OPERATORS....Pages 209-236
FUZZY APPROXIMATION BY FUZZY CONVOLUTION OPERATORS....Pages 237-261
DEGREE OF APPROXIMATION OF FUZZY NEURAL NETWORK OPERATORS, UNIVARIATE CASE....Pages 263-288
HIGHER DEGREE OF FUZZY APPROXIMATION BY FUZZY WAVELET TYPE AND NEURAL NETWORK OPERATORS....Pages 289-308
FUZZY RANDOM KOROVKIN THEOREMS AND INEQUALITIES....Pages 309-345
FUZZY-RANDOM NEURAL NETWORK APPROXIMATION OPERATORS, UNIVARIATE CASE....Pages 347-372
$\mathcal A$ -SUMMABILITY AND FUZZY KOROVKIN APPROXIMATION....Pages 373-384
$\mathcal A$ -SUMMABILITY AND FUZZY TRIGONOMETRIC KOROVKIN APPROXIMATION....Pages 385-397
UNIFORM REAL AND FUZZY ESTIMATES FOR DISTANCES BETWEEN WAVELET TYPE OPERATORS AT REAL AND FUZZY ENVIRONMENT....Pages 399-429
Back Matter....Pages -
