Ordinary and Fractional Approximation by Non-additive Integrals: Choquet, Shilkret and Sugeno Integral Approximators

355 Pages·2019·3.81 MB·English

by  George A. Anastassiou

Studies in Systems, Decision and Control 190
George A. Anastassiou
Ordinary and Fractional
Approximation by
Non-additive Integrals:
Choquet, Shilkret
and Sugeno Integral
Approximators
Studies in Systems, Decision and Control
Volume 190
Series editor
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George A. Anastassiou
Ordinary and Fractional
Approximation
by Non-additive Integrals:
Choquet, Shilkret and Sugeno
Integral Approximators
123
George A.Anastassiou
Department ofMathematical Sciences
University of Memphis
Memphis,TN, USA
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems,DecisionandControl
ISBN978-3-030-04286-8 ISBN978-3-030-04287-5 (eBook)
https://doi.org/10.1007/978-3-030-04287-5
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Preface
Nonlinear mathematics extend the linear mathematics to meet various needs and
demandsofpureandappliedmathematics,amongotherstocoveragreatvarietyof
applications in the real world. Ordinary and fractional approximations by
non-additiveintegrals,especiallybyintegralapproximatorsofChoquet,Silkretand
Sugeno types are a new trend in approximation theory. These integrals are only
subadditiveandonlythefirsttwoarepositivelinear,andtheyproduceveryfastand
flexible approximations based on limited data. We present both the univariate and
multivariate cases. The Choquet integral is based on Choquet capacities, the
Shilkret integral is based on maxitive measures and the Sugeno fuzzy integral is
based on fuzzy measures. All these set functions are much weaker forms of the
Lebesguemeasureandtheywereconceivedtofulfilneedsofeconomictheoriesand
other applied sciences. This monograph is the natural evolution of recent author’s
research work put in a book form for the first time. The presented approaches are
original, and chapters are self-contained and can be read independently. This
monograph is suitable to be used in related graduate classes and research projects.
We exhibit to the maximum our approximation methods to all possible directions.
For the history, G. Choquet (1953) introduced the capacities and his integral.
Initially these were applied to statistical mechanics and potential theory, and they
gave rise to the study of non-additive measure theory. Slowly but steadily these
ideas of Choquet started to attract economists especially after the very important
workofShapley(1953)inthestudyofcooperativegames.CapacitiesandChoquet
integralsbecamemainstreamtoDecisiontheoristssince1989whenD.Schmeidler
was the first to use them in a model of choice with non-additive beliefs. The
expectedutilityresultsarestrengthenedbytheuseofChoquetcapacitiesinsteadof
probability measures. Nowadays, Choquet integral has wide applications, among
others, to decision making under risk and uncertainty, in finance, in economics, in
portfolioproblemsandininsurance.Themotivationtowritethismonographcame
also by the following: various issues related to the modeling and analysis of
ordinary and fractional order systems have gained an increased popularity, as
witnessedbymanybooksandvolumesinSpringer’sprogram:http://www.springer.
com/gp/search?query=fractional&submit=Prze%C5%9Blij and the purpose of our
ix
x Preface
bookistoprovideadeeperformalanalysisonsomeissuesthatarerelevanttomany
areas for instance: decision making, complex processes, systems modeling and
control and related areas. The above are deeply embedded in the fields of mathe-
matics,engineering,computerscience,physics,economics,socialandlifesciences.
The list of presented topics are as follows:
(cid:129) QuantitativeApproximationbyKantorovich-Choquetquasi-interpolationneural
network operators
(cid:129) QuantitativeApproximationbyPerturbedKantorovich-ChoquetNeuralNetwork
Operators
(cid:129) Approximation by Shift Invariant Univariate Sublinear-Choquet Operators
(cid:129) Quantitative Approximation by Shift Invariant Multivariate Sublinear-Choquet
Operators
(cid:129) Hardy type inequalities for Choquet integrals
(cid:129) Approximation by Choquet integrals
(cid:129) Conformable Fractional Approximation by Choquet integrals
(cid:129) Multivariate and Convex Approximation by Choquet integrals
(cid:129) Caputo and Canavati fractional Approximation by Choquet integrals
(cid:129) MixedConformableandIteratedfractionalApproximationbyChoquetintegrals
(cid:129) Quantitative Approximation by a Kantorovich-Shilkret quasi-interpolation neu-
ral networkoperator
(cid:129) Multivariate Approximation with Rates by Perturbed Kantorovich-Shilkret
Neural Network Operators
(cid:129) Approximation by Shift Invariant Univariate Sublinear-Shilkret Operators
(cid:129) Approximation by Shift Invariant Multivariate Sublinear-Shilkret Operators
(cid:129) Ostrowski-Sugeno Fuzzy inequalities
(cid:129) Fractional Ostrowski-Sugeno Fuzzy univariate Inequalities
(cid:129) Multivariate Ostrowski-Sugeno Fuzzy Inequalities
An extensive list of references is given per chapter.
The book’s results are expected to find applications in many areas of pure and
applied mathematics, especially in approximation theory, numerical analysis and
mathematical economics in both ordinary and fractional sense. As such this
monographissuitableforresearchers,graduatestudentsandseminarsoftheabove
disciplines, also to be in all science and engineering libraries.
The preparation of the book took place during 2017–2018 at the University of
Memphis.
The author likes to thank Prof. Alina Lupas of University of Oradea, Romania,
for checking and reading the manuscript.
Memphis, TN, USA George A. Anastassiou
September 2018
Contents
1 Approximation with Rates by Kantorovich–Choquet
Quasi-interpolation Neural Network Operators . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Results - I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Results - II. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Approximation with Rates by Perturbed Kantorovich–Choquet
Neural Network Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Univariate Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 Multivariate Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Approximation with Rates by Shift Invariant Univariate
Sublinear-Choquet Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Univariate Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Higher Order of Approximation. . . . . . . . . . . . . . . . . . . . . . . . 65
3.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4 Approximation with Rates by Shift Invariant Multivariate
Sublinear-Choquet Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Multivariate Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Higher Order Multivariate Approximation . . . . . . . . . . . . . . . . 84
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