Mathematical Analysis and Applications E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
An authoritative text that presents the current problems, theories, and applications of mathematical analysis research Mathematical Analysis and Applications: Selected Topics offers the theories, methods, and applications of a variety of targeted topics including: operator theory, approximation theory, fixed point theory, stability theory, minimization problems, many-body wave scattering problems, Basel problem, Corona problem, inequalities, generalized normed spaces, variations of functions and sequences, analytic generalizations of the Catalan, Fuss, and Fuss-Catalan Numbers, asymptotically developable functions, convex functions, Gaussian processes, image analysis, and spectral analysis and spectral synthesis. The authors--a noted team of international researchers in the field-- highlight the basic developments for each topic presented and explore the most recent advances made in their area of study. The text is presented in such a way that enables the reader to follow subsequent studies in a burgeoning field of research. This important text: * Presents a wide-range of important topics having current research importance and interdisciplinary applications such as game theory, image processing, creation of materials with a desired refraction coefficient, etc. * Contains chapters written by a group of esteemed researchers in mathematical analysis Includes problems and research questions in order to enhance understanding of the information provided * Offers references that help readers advance to further study Written for researchers, graduate students, educators, and practitioners with an interest in mathematical analysis, Mathematical Analysis and Applications: Selected Topics includes the most recent research from a range of mathematical fields.
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Table of Contents
Cover
Title Page
Copyright
Preface
About the Editors
List of Contributors
Chapter 1: Spaces of Asymptotically Developable Functions and Applications
1.1 Introduction and Some Notations
1.2 Strong Asymptotic Expansions
1.3 Monomial Asymptotic Expansions
1.4 Monomial Summability for Singularly Perturbed Differential Equations
1.5 Pfaffian Systems
References
Chapter 2: Duality for Gaussian Processes from Random Signed Measures
2.1 Introduction
2.2 Reproducing Kernel Hilbert Spaces (RKHSs) in the Measurable Category
2.3 Applications to Gaussian Processes
2.4 Choice of Probability Space
2.5 A Duality
2.A Stochastic Processes
2.B Overview of Applications of RKHSs
Acknowledgments
References
Chapter 3: Many-Body Wave Scattering Problems for Small Scatterers and Creating Materials with a Desired Refraction Coefficient
3.1 Introduction
3.2 Derivation of the Formulas for One-Body Wave Scattering Problems
3.3 Many-Body Scattering Problem
3.4 Creating Materials with a Desired Refraction Coefficient
3.5 Scattering by Small Particles Embedded in an Inhomogeneous Medium
3.6 Conclusions
References
Chapter 4: Generalized Convex Functions and their Applications
4.1 Brief Introduction
4.2 Generalized E-Convex Functions
4.3 -Epigraph
4.4 Generalized -Convex Functions
4.5 Applications to Special Means
References
Chapter 5: Some Properties and Generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers
5.1 The Catalan Numbers
5.2 The Catalan–Qi Function
5.3 The Fuss–Catalan Numbers
5.4 The Fuss–Catalan–Qi Function
5.5 Some Properties for Ratios of Two Gamma Functions
5.6 Some New Results on the Catalan Numbers
5.7 Open Problems
Acknowledgments
References
Chapter 6: Trace Inequalities of Jensen Type for Self-adjoint Operators in Hilbert Spaces: A Survey of Recent Results
6.1 Introduction
6.2 Jensen's Type Trace Inequalities
6.3 Reverses of Jensen's Trace Inequality
6.4 Slater's Type Trace Inequalities
References
Chapter 7: Spectral Synthesis and Its Applications
7.1 Introduction
7.2 Basic Concepts and Function Classes
7.3 Discrete Spectral Synthesis
7.4 Nondiscrete Spectral Synthesis
7.5 Spherical Spectral Synthesis
7.6 Spectral Synthesis on Hypergroups
7.7 Applications
Acknowledgments
References
Chapter 8: Various Ulam–Hyers Stabilities of Euler–Lagrange–Jensen General (a,b;k=a+b)-Sextic Functional Equations
8.1 Brief Introduction
8.2 General Solution of Euler–Lagrange–Jensen General -Sextic Functional Equation
8.3 Stability Results in Banach Space
8.4 Stability Results in Felbin's Type Spaces
8.5 Intuitionistic Fuzzy Normed Space: Stability Results
References
Chapter 9: A Note on the Split Common Fixed Point Problem and its Variant Forms
9.1 Introduction
9.2 Basic Concepts and Definitions
9.3 A Note on the Split Equality Fixed-Point Problems in Hilbert Spaces
9.4 Numerical Example
9.5 The Split Feasibility and Fixed Point Problems for Quasi-Nonexpansive Mappings in Hilbert Spaces
9.6 Ishikawa-Type Extra-Gradient Iterative Methods for Quasi-Nonexpansive Mappings in Hilbert Spaces
9.7 Conclusion
References
Chapter 10: Stabilities and Instabilities of Rational Functional Equations and Euler–Lagrange–Jensen (a,b)-Sextic Functional Equations
10.1 Introduction
10.2 Ulam Stability Problem for Functional Equation
10.3 Various Forms of Functional Equations
10.4 Preliminaries
10.5 Rational Functional Equations
10.6 Euler-Lagrange–Jensen -Sextic Functional Equations
References
Chapter 11: Attractor of the Generalized Contractive Iterated Function System
11.1 Iterated Function System
11.2 Generalized -contractive Iterated Function System
11.3 Iterated Function System in -Metric Space
11.4 Generalized -Contractive Iterated Function System in -Metric Space
References
Chapter 12: Regular and Rapid Variations and Some Applications
12.1 Introduction and Historical Background
12.2 Regular Variation
12.3 Rapid Variation
12.4 Applications to Selection Principles
12.5 Applications to Differential Equations
References
Chapter 13:
n
-Inner Products,
n
-Norms, and Angles Between Two Subspaces
13.1 Introduction
13.2 -Inner Product Spaces and -Normed Spaces
13.3 Orthogonality in -Normed Spaces
13.4 Angles Between Two Subspaces
References
Chapter 14: Proximal Fiber Bundles on Nerve Complexes
14.1 Brief Introduction
14.2 Preliminaries
14.3 Sewing Regions Together
14.4 Some Results for Fiber Bundles
14.5 Concluding Remarks
References
Chapter 15: Approximation by Generalizations of Hybrid Baskakov Type Operators Preserving Exponential Functions
15.1 Introduction
15.2 Baskakov–Szász Operators
15.3 Genuine Baskakov–Szász Operators
15.4 Preservation of
15.5 Conclusion
References
Chapter 16: Well-Posed Minimization Problems via the Theory of Measures of Noncompactness
16.1 Introduction
16.2 Minimization Problems and Their Well-Posedness in the Classical Sense
16.3 Measures of Noncompactness
16.4 Well-Posed Minimization Problems with Respect to Measures of Noncompactness
16.5 Minimization Problems for Functionals Defined in Banach Sequence Spaces
16.6 Minimization Problems for Functionals Defined in the Classical Space
16.7 Minimization Problems for Functionals Defined in the Space of Functions Continuous and Bounded on the Real Half-Axis
References
Chapter 17: Some Recent Developments on Fixed Point Theory in Generalized Metric Spaces
17.1 Brief Introduction
17.2 Some Basic Notions and Notations
17.3 Fixed Points Theorems
17.4 Common Fixed Points Theorems
17.5 Best Proximity Points
17.6 Common Best Proximity Points
17.7 Tripled Best Proximity Points
17.8 Future Works
References
Chapter 18: The Basel Problem with an Extension
18.1 The Basel Problem
18.2 An Euler Type Sum
18.3 The Main Theorem
18.4 Conclusion
References
Chapter 19: Coupled Fixed Points and Coupled Coincidence Points via Fixed Point Theory
19.1 Introduction and Preliminaries
19.2 Fixed Point Results
19.3 Coupled Fixed Point Results
19.4 Coincidence Point Results
19.5 Coupled Coincidence Results
References
Chapter 20: The Corona Problem, Carleson Measures, and Applications
20.1 The Corona Problem
20.2 Carleson's Proof and Carleson Measures
20.3 The Corona Problem in Higher Henerality
20.4 Results on Carleson Measures
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Preface
Begin Reading
List of Illustrations
Chapter 2: Duality for Gaussian Processes from Random Signed Measures
Figure 2.1 The sequence in s.t. , as .
Figure 2.2 The RKHSs , .
Figure 2.A.1 Gaussian distribution , (variance).
Figure 2.A.2 Monte-Carlo simulation of Brownian motion starting at , with five sample paths. (“Monte-Carlo simulation” refers to the use of computer-generated random numbers.)
Figure 2.B.1 and its distributional derivative.
Figure 2.B.2 Examples of configuration of resistors in a network.
Figure 2.B.3 Covariance between vertices.
Figure 2.B.5 A binary tree model.
Chapter 9: A Note on the Split Common Fixed Point Problem and its Variant Forms
Figure 9.1 Shows the convergence of Example 9.80 Algorithm (9.99), starting with the initial value and
Figure 9.2 Shows the convergence of Example 9.80 Algorithm (9.99), starting with the initial value and .
Figure 9.3 Shows the convergence of Example 9.7 Algorithm (9.101), starting with the initial value and
Figure 9.4 Shows the convergence of Example 9.7 Algorithm (9.101), starting with the initial value and
Chapter 10: Stabilities and Instabilities of Rational Functional Equations and Euler–Lagrange–Jensen (a,b)-Sextic Functional Equations
Figure 10.1 Geometrical interpretation of (10.41).
Figure 10.2 Electric circuit connected with two resistors in parallel.
Chapter 11: Attractor of the Generalized Contractive Iterated Function System
Figure 11.1 Convergence of to the attractor of IFS.
Chapter 14: Proximal Fiber Bundles on Nerve Complexes
Figure 14.1 Two forms of fiber bundles: spatial: and descriptive: .
Figure 14.2 Nrv
K
.
Figure 14.3 Two forms of nerve 2-spokes.
Figure 14.4 .
Figure 14.5 Strongly near filled triangles.
Figure 14.7 Strongly near.
Figure 14.6 .
Figure 14.8 Sewing disconnected regions together.
Figure 14.9 Sewing nerve complexes together.
Figure 14.10 Two fiber bundles containing projections on a pair of nerve complexes to a pair of descriptions of the nerves and on a nervous system to a description of the nervous system.
Figure 14.11 Two fiber bundles containing projections on a pair of nerve complexes to a pair of descriptions of the nerves and on a nervous system to a description of the nervous system.
List of Tables
Chapter 9: A Note on the Split Common Fixed Point Problem and its Variant Forms
Table 9.1 Shows the numerical values of Example 9.80 Algorithm (9.99), starting with the initial values and
Table 9.2 Shows the numerical values of Example 9.80 Algorithm (9.99), starting with the initial values and
Table 9.3 Shows the numerical values of Example 9.7 Algorithm (9.101), starting with the initial values and
Table 9.4 Shows the numerical values of Example 9.7 Algorithm (9.101), starting with the initial values and
Mathematical Analysis and Applications
Selected Topics
Edited by
Michael Ruzhansky
Hemen Dutta
Ravi P. Agarwal
This edition first published 2018
© 2018 John Wiley & Sons, Inc.
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Library of Congress Cataloging-in-Publication Data:
Names: Ruzhansky, M. (Michael), editor. | Dutta, Hemen, 1981- editor. | Agarwal, Ravi P., editor.
Title: Mathematical analysis and applications : selected topics / edited by Michael Ruzhansky, Hemen Dutta, Ravi P. Agarwal.
Description: Hoboken, NJ : John Wiley & Sons, 2018. | Includes bibliographical references and index. |
Identifiers: LCCN 2017048922 (print) | LCCN 2017054738 (ebook) | ISBN 9781119414308 (pdf) | ISBN 9781119414339 (epub) | ISBN 9781119414346 (cloth)
Subjects: LCSH: Mathematical analysis.
Classification: LCC QA300 (ebook) | LCC QA300 .M225 2018 (print) | DDC 515-dc23
LC record available at https://lccn.loc.gov/2017048922
Cover Design: Wiley
Cover Image: © LoudRedCreative/Getty Images
Preface
This book is designed for researchers, graduate students, educators, and practitioners with an interest in mathematical analysis in particular and in mathematics in general. The book aims to present theory, methods, and applications of the chosen topics under several chapters that have recent research importance and use. Emphasis is made to present the basic developments concerning each idea in full detail, and also contain the most recent advances made in the corresponding area of study. The text is presented in a self-contained manner, providing at least an idea of the proof of all results, and giving enough references to enable the interested reader to follow subsequent studies in a still developing field. There are 20 selected chapters in this book and they are organized as follows.
The chapter “Spaces of Asymptotically Developable Functions and Applications” presents the functional structure of the spaces of asymptotically developable functions in several complex variables. The authors also illustrate the notion of summability with some applications concerning singularly perturbed systems of ordinary differential equations and Pfaffian systems.
The chapter “Duality for Gaussian Processes from Random Signed Measures” proves a number of results for a general class of Gaussian processes. Two features are stressed, first the Gaussian processes are indexed by a general measure space; second, the authors “adjust” the associated reproducing kernel Hilbert spaces (RKHSs) to the measurable category. Among other things, this allows us to give a precise necessary and sufficient condition for equivalence of a pair of probability measures (in sample space), which determine the corresponding two Gaussian processes.
In the chapter “Many-body Wave Scattering Problems for Small Scatterers and Creating Materials with a Desired Refraction Coefficient,” formulas are derived for solutions of many-body wave scattering problems by small impedance particles embedded in an inhomogeneous medium. The limiting case is considered when the size a of small particles tends to zero while their number tends to infinity at a suitable rate. Equations for the limiting effective (self-consistent) field in the medium are derived. The theory is based on a study of integral equations and asymptotic of their solutions as a tends to zero. The case of wave scattering by many small particles embedded in an inhomogeneous medium is also studied. Applications of this theory to creating materials with a desired refraction coefficient are given. A recipe is given for creating such materials by embedding into a given material many small impedance particles with prescribed boundary impedances.
The chapter “Generalized Convex Functions and their Applications” focuses on convex functions and their generalization. The definition of convex function along with some relevant properties of such functions is given first, followed by a discussion on a simple geometric property. Then the e-convex function is generalized and some of their properties are established. Moreover, a generalized s-convex function is presented in the second sense and the paper presents some new inequalities of generalized Hermite–Hadamard's type for the class of functions whose second local fractional derivatives of order α in absolute value at certain powers are generalized s-convex functions in the second sense. At the end, some applications to special means are also presented.
The chapter “Some Properties and Generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers” presents an expository review and survey for analytic generalizations and properties of the Catalan numbers, the Fuss numbers, the Fuss–Catalan numbers, the Catalan function, the Catalan–Qi function, the q-Catalan–Qi numbers, and the Fuss–Catalan–Qi function.
The chapter “Trace Inequalities of Jensen Type for Selfadjoint Operators in Hilbert Spaces: A Survey of Recent Results” provides a survey of recent results for trace inequalities related to the celebrated Jensen's and Slater's inequalities and their reverses. Applications for various functions of interest such as power and logarithmic functions are also emphasized. Trace inequalities for bounded linear operators in complex Hilbert spaces play an important role in Physics, in general, and in Quantum Mechanics, in particular.
The chapter “Spectral Synthesis and its Applications” presents a survey about spectral analysis and spectral synthesis. The chapter recalls the most important classical results in the field and in some cases new proofs for them are given. It also presents the most recent results in discrete, nondiscrete, and spherical spectral synthesis together with some applications.
The chapter “Various Ulam–Hyers Stabilities of Euler–Lagrange–Jensen General (a, b; k = a + b)-Sextic Functional Equations” elucidates the historical development of well-known stabilities of various types of functional equations such as quintic, sextic, septic, and octic functional equations. It introduced a new generalized Euler–Lagrange–Jensen sextic functional equation, obtained its general solution and further investigated its various fundamental stabilities and instabilities by having employed the famous Hyers' direct method as well as the alternative fixed point method. The chapter is expected to be helpful to analyze the stability of various functional equations applied in the physical sciences.
The chapter “A Note on the Split Common Fixed-Point Problem and its Variant Forms” proposed new algorithms for solving the split common fixed point problem and its variant forms, and prove the convergence results of the proposed algorithms. The split common fixed point problems have found its applications in various branches of mathematics both pure and applied. It provides a unified structure to study a large number of nonlinear mappings.
The chapter “Stabilities and Instabilities of Rational Functional Equations and Euler–Lagrange–Jensen (a, b)-Sextic Functional Equations” comprises the growth, importance and relevance of functional equations in other fields. Its fundamental and basic results of various stabilities are presented. The stability results of various rational and Euler–Lagrange–Jensen sextic functional equations are investigated. Application and geometrical interpretation of rational functional equation are also illustrated for the readers to study similar problems.
The chapter “Attractor of the Generalized Contractive Iterated Function System” deals with the problems to construct the fractal sets of the iterated function system for certain finite collection of F-contraction mappings defined on metric spaces as well as b-metric spaces. A new iterated function system called generalized F-contractive iterated function system is defined. Further, a method is presented to construct new fractals; where the resulting fractals are often self-similar but more general.
The chapter “Regular and Rapid Variations and some Applications” presents an overview of recent results on regular and rapid variations of functions and sequences and some their applications in selection principles theory, game theory, and asymptotic analysis of solutions of differential equations.
The chapter “n-Inner Products, n-Norms, and Angles Between Two Subspaces” discusses the concepts of n-inner products and n-norms for any natural number n, which are generalizations of the concepts of inner products and norms. It presents some geometric aspects of n-normed spaces and n-inner product spaces, especially regarding the notion of orthogonality and angles between two subspaces of such a space.
The chapter “Proximal Fiber Bundles on Nerve Complexes” introduces proximal fiber bundles of nerve complexes. Briefly, a nerve complex is a collection of filled triangles (2-simplexes) that have nonempty intersection. Nerve complexes are important in the study of shapes with a number of recent applications that include the classification of object shapes in digital images. The focus of this chapter is on fiber bundles defined by projections on a set of fibers that are nerve complexes into a base space such as the set of descriptions of nerve complexes. Two forms of fiber bundles are introduced, namely, spatial and descriptive, including a descriptive form BreMiller–Sloyer sheaf on a Vietoris–Rips complex. The introduction to nerve complexes includes a recent extension of nerve complexes that includes nerve spokes. A nerve spoke is a collection of filled triangles that always includes filled triangle in a nerve complex. A natural transition from the study of fibers that are nerve complexes is in the form of projections of pairs of fibers onto a local nervous system complex, which is a collection of nerve complexes that are glued together with spokes common to the nerve fibers. A number of results are given for fiber bundles viewed in the context of proximity spaces.
The chapter “Approximation by Generalizations of Hybrid Baskakov Type Operators Preserving Exponential Functions” deals with the approximation properties of the certain Baskakov–Szász operators. It estimates the results for these hybrid Baskakov–Szász type operators for exponential test functions. It also estimates a quantitative asymptotic formula for such operators. Mathematica software is used to estimate the results.
The chapter “Well-Posed Minimization Problems via the Theory of Measures of Noncompactness” presents an analysis of the minimization problems for functionals defined, lower bounded and lower semicontinuous on a closed subset of a metric space. The main focus is on the well-posedness of minimization problems from the viewpoint of the theory of measures of noncompactness. The minimization problems for several functionals defined on some Banach spaces are also investigated as well. Thus, the chapter clarifies the role of the theory of measures of noncompactness in the general approach to the well-posedness of minimization problems.
The chapter “Some Recent Developments on Fixed Point Theory in Generalized Metric Spaces” discusses some important developments in the fixed point theory in metric spaces. Various advancements are explained in detail through useful and applicable results along with examples in generalized metric spaces.
The chapter “The Basel Problem with an Extension” presents some historical aspects to the famous Basel problem, which a number of brilliant mathematicians attempted, and which had remained unsolved for over 90 years. It was the genius Euler who provided a masterful solution and laid the foundations to the famous Riemann zeta function and the analysis of series. The chapter then investigates a related Euler sum and provides an explicit analytical representation, a closed form solution. The related Euler sum also represented in terms of logarithmic and hypergeometric functions. The integrals in question will be associated with the harmonic numbers of positive terms. A few examples of integrals provide an identity in terms of some special functions.
The chapter “Coupled Fixed Points and Coupled Coincidence Points via Fixed Point Theory” focuses on the study of the coupled fixed point and coupled coincidence point problems for single- and multi-valued operators. The study of this chapter is based on appropriate fixed point theorems in two types of generalized metric spaces. Some applications are also given to illustrate part of the abstract results presented in this chapter.
The chapter “The Corona Problem, Carleson Measures, and Applications” reviews the developments and generalizations of the Corona problem, the results on Carleson measures themselves and some applications of Carleson measures, in several different settings, starting from the disc in ℂ (where the corona problem was originally set) arriving to the unit ball in ℂn, to bounded strongly pseudoconvex domains and even to domains in the quaternionic space. Both the corona problem and the Carleson measures still need investigation, as many open problems have not been solved yet. The open problems are also highlighted in this chapter. Carleson measures were introduced by Lennart Carleson in 1962 to solve an interpolation problem about bounded holomorphic functions called the corona problem.
The editors are grateful to the contributors for their timely contribution and co-operation throughout the editing process. The editors have benefited from the remarks and comments of several other experts on the topics covered in this book. The editors would also like to thank the book handling editors at Wiley and production staff members for their continuous support and help. Finally, the editors offer sincere thanks to all those who contributed directly or indirectly in completing this book project.
August 25, 2017
Michael Ruzhansky London, UKHemen Dutta Guwahati, IndiaRavi P. Agarwal Texas, USA
About the Editors
Michael Ruzhansky is a Professor at the Department of Mathematics, Imperial College London, UK. He has published over 100 research articles in leading international journals. He has also published 5 books and memoirs, and 9 edited volumes. His major research topics are related to pseudo-differential operators, harmonic analysis, functional analysis, partial differential equations, boundary value problems, and their applications. He is serving on the editorial board of many respected international journals and served as the President of the International Society of Analysis, Applications, and Computations (ISAAC) in the period 2009–2013.
Hemen Dutta is a Senior Assistant Professor of Mathematics at Gauhati University, India. He did his Master of Science (M.Sc.) in Mathematics, Post Graduate Diploma in Computer Application (PGDCA) and Ph.D. in Mathematics from Gauhati University, India. He received his M.Phil. in Mathematics from Madurai Kamaraj University, India. His research interest includes summability theory and functional analysis. He has to his credit more than 50 research papers and three books so far. He has delivered talks at foreign and national institutions. He has also organized a number of academic events. He is a member of several mathematical societies.
Ravi P. Agarwal is a Professor and the chair of the Department of Mathematics at Texas A&M University-Kingsville, USA. He has been actively involved in research as well as pedagogical activities for the last 45 years. Dr. Agarwal is the author or co-author of more than 1400 scientific papers and 40 research monographs. His major research interests include numerical analysis, differential and difference equations, inequalities, and fixed point theorems. Dr. Agarwal is the recipient of several notable honors and awards. He is on the editorial board of several journals in different capacities and also organized International Conferences.
List of Contributors
Mujahid Abbas
Department of Mathematics
Government College University
Katchery Road, Lahore 54000
Pakistan
and
Department of Mathematics
King Abdulaziz University
Jeddah 21589
Saudi Arabia
Józef Banaś
Department of Nonlinear Analysis
Rzeszów University of Technology
Aleja Powstańców Warszawy 8 35-959 Rzeszów
Poland
Dragan Djurčić
Faculty of Technical Sciences Department of Mathematics
University of Kragujevac
34000 Čačak
Serbia
Silvestru Sever Dragomir
Mathematics Department College of Engineering & Science
Victoria University
Melbourne 8001
Australia
and
DST-NRF Centre of Excellence in the Mathematical and Statistical Sciences
School of Computer Science and Applied Mathematics
University of the Witwatersrand
Johannesburg 2000
South Africa
Jorge Mozo Fernández
Departamento de Álgebra, Análisis Matemático Geometría y Topología Facultad de Ciencias
Campus Miguel Delibes
Universidad de Valladolid
Paseo de Belén, 7, 47011 Valladolid
Spain
Hendra Gunawan
Department of Mathematics
Bandung Institute of Technology
Bandung 40132
Indonesia
Bai-Ni Guo
School of Mathematics and Informatics
Henan Polytechnic University
Jiaozuo, Henan, 454010
China
Vijay Gupta
Department of Mathematics
Netaji Subhas Institute of Technology
Dwarka, New Delhi 110078
India
Palle E.T. Jorgensen
Department of Mathematics
The University of Iowa
Iowa City, IA 52242-1419
USA
Adem Kiliçman
Department of Mathematics Faculty of Science Putra University of Malaysia
43400 Serdang, Selangor
Malaysia
Ljubiša D.R. Kočinac
Faculty of Sciences and Mathematics Department of Mathematics
University of Niš, 18000 Niš
Serbia
Somayya Komal
Theoretical and Computational Science (TaCS) Centre, Department of Mathematics, Faculty of Science
King Mongkut's University of Technology Thonburi
Thung Khru, Bangkok 10140
Thailand
Poom Kumam
Theoretical and Computational Science (TaCS) Centre, Department of Mathematics, Faculty of Science
King Mongkut's University of Technology Thonburi
Thung Khru, Bangkok 10140
Thailand
Beri V. Senthil Kumar
Department of Mathematics
C. Abdul Hakeem College of Engg. and Tech.
Melvisharam 632 509, Tamil Nadu
India
Jelena V. Manojlović
Faculty of Sciences and Mathematics Department of Mathematics
University of Niš
18000 Niš
Serbia
L.B. Mohammed
Department of Mathematics Faculty of Science
Universiti Putra Malaysia
43400 Serdang, Selangor
Malaysia
Talat Nazir
Department of Mathematics
University of Jeddah
Jeddah 21589
Saudi Arabia
and
Department of Mathematics
COMSATS Institute of Information Technology
Abbottabad 22060
Pakistan
Narasimman Pasupathi
Department of Mathematics
Thiruvalluvar University College of Arts and Science
Tirupattur 635 901, Tamil Nadu
India
James F. Peters
Computational Intelligence Laboratory
University of Manitoba
WPG, MB, R3T 5V6
Canada
and
Department of Mathematics Faculty of Arts and Sciences
Adiyaman University
02040 Adiyaman
Turkey
Adrian Petruşel
Faculty of Mathematics and Computer Science
Babeş-Bolyai University
400084 Cluj-Napoca
Romania
Gabriela Petruşel
Faculty of Business
Babeş-Bolyai University
400084 Cluj-Napoca
Romania
Feng Qi
Institute of Mathematics
Henan Polytechnic University
Jiaozuo, Henan, 454010
China
and
College of Mathematics
Inner Mongolia University for Nationalities
Tongliao, Inner Mongolia, 028043
China
and
Department of Mathematics College of Science
Tianjin Polytechnic University
Tianjin, 300387
China
Alexander G. Ramm
Department of Mathematics
Kansas State University
Manhattan, KS 66506-2602
USA
John Michael Rassias
Pedagogical Department E.E., Section of Mathematics and Informatics
National and Capodistrian University of Athens
Athens 15342
Greece
Krishnan Ravi
Department of Mathematics
Sacred Heart College
Tirupattur 635 601, Tamil Nadu
India
Wedad Saleh
Department of Mathematics, Faculty of Science
Putra University of Malaysia
43400 Serdang, Selangor
Malaysia
Alberto Saracco
Dipartimento di Scienze Matematiche, Fisiche e Informatiche
Universitá degli Studi di Parma, 43124
Italy
Anthony Sofo
Victoria University
Melbourne City, Victoria 8001
Australia
László Székelyhidi
Institute of Mathematics
University of Debrecen, H-4010 Debrecen
Hungary
and
Department of Mathematics
University of Botswana
Gaborone
Botswana
Feng Tian
Department of Mathematics
Hampton University
Hampton, VA 23668
USA
Sergio Alejandro Carrillo Torres
Escuela de Ciencias Exactas e Ingeniería
Universidad Sergio Arboleda
Calle 74, 14-14, Bogotá
Colombia
Tomasz Zając
Department of Nonlinear Analysis
Rzeszów University of Technology
Aleja Powstańców Warszawy 8 35-959 Rzeszów
Poland
Chapter 1Spaces of Asymptotically Developable Functions and Applications
Sergio Alejandro Carrillo Torres1 and Jorge Mozo Fernández2
1Escuela de Ciencias Exactas e Ingeniería, Universidad Sergio Arboleda, Calle 74, 14-14, Bogotá, Colombia
2Departamento de Álgebra, Análisis Matemático, Geometría y Topología, Facultad de Ciencias, Campus Miguel Delibes, Universidad de Valladolid, Paseo de Belén, 7, 47011 Valladolid, Spain
2010 AMS Subject Classification Primary 34E05, 34E15
1.1 Introduction and Some Notations
This chapter is a short review of some results concerning asymptotic expansions in several complex variables, and summability. Different notions of asymptotic expansions have been developed in literature in last decades, trying to generalize classical results regarding asymptotics in one variable, dating back to H. Poincaré, and further developed by Wasow [1], Ramis [2], Écalle [3], Balser [4], Braaksma [5, 6], and many others. In several variables, as main contributions, let us mention those of Gérard and Sibuya [7] in the 1970s, Majima [8, 9] in the 1980s, and more recently the notion of monomial asymptotic expansions and monomial summability of Canalis-Durand et al. [10]. It is also worth to mention here that the notion of composite asymptotic expansions, developed by Fruchard and Schäfke [11], has been very useful in the treatment of singularly perturbed linear differential equations.
We shall review mainly the notion of strong asymptotic expansions of Majima and monomial asymptotic expansions of Canalis-Durand, Mozo, and Schäfke, and clarifying some relations between them. These notions will be applied to several problems concerning summability of solutions of systems of ordinary differential equations ODEs and Pfaffian systems. Concerning Pfaffian systems, we will state some recent progresses of S. Carrillo, see [12] for complete details.
This chapter does not intend to be complete at all, the objective is only to present, in the opinion of the authors, some of the more relevant contributions concerning this wide theory. For the new results presented here, relevant precise references are given in the text. Such a survey, as we know, does not exist in the literature, so we think that it may be useful as a starting point for researchers in the area.
Some of the main notations used throughout the text will be the following:
will denote the set of natural numbers including 0, and
.
,
,
,…(boldface) will denote vectors:
, and so on.
If
,
,
means that
,
.
If
,
,
means that
,
.
.
If is an open set in , is the set of holomorphic functions on . Similarly, will denote the set of functions on , identifying .
is the ring of formal power series in the variables .
is the ring of convergent power series (at the origin) in the variables .
If and (or ), will denote the sector of radius and opening between the rays and , that is, the set
Note that we are not restricted to the case . If , the sectors are to be understood in the Riemann surface of , that is, the universal covering of .
If and , will denote the polysector
In general, multiindex notation will be freely used throughout the text. So, will denote , is , and so on.
1.2 Strong Asymptotic Expansions
The notion of asymptotics in one variable was introduced by Poincaré, trying to give a meaning to the notion of a sum for divergent series, that had been controversed and widely study since the times of Euler and Abel and was developed by different authors during the twentieth century, as Birkhoff, Wasow, Hukuhara, and Sibuya. Some good expositions with emphasis in the history, of the theory of divergent series, are due to Ramis [13, 14]. An important improvement was done at the end of the 1970s by Ramis with the introduction of the theory of Gevrey summability, generalizing Borel summability. The objective was to give a meaning to the formal power series appearing as solutions of systems of ODEs with irregular singular points. In other words, to define a sum (or several sums) for these series. It turns out that not every solution of a system of ODEs with irregular singular points is summable in this sense, but nevertheless it was shown in the next years that solutions of these systems are multisummable, that is, choosing a direction in the complex plane that avoids a finite number of directions, the formal series solutions can be uniquely decomposed as a sum of formal series, and a process of -summability (with different values of ) can be applied to each of these summands in order to obtain a true holomorphic solution of the system.
Different essays were done in order to generalize this notion to several variables. Asymptotics and summability with parameters were used by different authors, from Wasow, Hukuhara, and others, but this was not a true complete notion of summability in several variables, that could be used, for instance, to study systems of partial differential equations. The first notion that clearly generalized that of Poincaré was due to H. Majima, who in 1983 presented what he called strong asymptotic expansions, and applied in 1984 to the study of integrable connections.
In this section, we shall recall the notion of strong asymptotic expansions. In the Gevrey case, his work was generalized by Haraoka [15]. Further developments of this notion were established by Zurro [16], Hernández [17], Galindo and Sanz [18] and the second author, among others.
Given , and , denote the polysector .
Definition 1.1
A total family of coefficients in is a family of holomorphic functions
Given such a family, let us define, for , the -approximant of as
where denotes , .
Definition 1.2
Let be a polysector in , , and a total family of coefficients in . We will say that admits as a strong asymptotic expansion in if for every , and every strict subpolysector of (see Remark 1.1), there are constants such that, if ,
If , the asymptotic expansion is called of -Gevrey type if there exists constants and , depending on the subpolysector , such that can be chosen as
Remark 1.1
In this remark, and throughout the text, we will say that is a strict subpolysector of if it is bounded, and moreover,
We will denote this situation as .
Let us denote (resp. ) the set of functions on admitting a strong asymptotic expansion (resp. of -Gevrey type). They are differential -algebras. In particular, if and admits a family as a strong asymptotic expansion, the derivative admits , where
as strong asymptotic expansion. The unicity of the asymptotic expansion can be deduced from the following fact: take, for instance, , . Then, we have that
in a proper subpolysector. Taking limits when tends to 0, we have
Stability under derivation allows us to conclude. Due to this unicity, the total family of coefficients of strong asymptotic expansion for a function is denoted as . Denoting , the formal power series
is the formal power series of asymptotic expansion of , denoted as . In the particular case, this series, when expanded with respect to any variable, has coefficients holomorphic in a common disk around the origin in the other variables, then it determines the family , and we will say that has the series as strong asymptotic expansion at the origin.
Definition 1.3
A total family of coefficients in is called consistent if each and moreover the family
equals . If , the family is a consistent one.
The following characterization turns out to be very useful in order to work with strong asymptotic expansions in polysectors:
Theorem 1.1
Let be a polysector in and . The following conditions are equivalent:
1.
.
2.
If
, every derivative
is bounded in
.
3.
If
, the restriction
can be extended to the whole space
as a
-function (considering
).
Proof
(1) (2) is evident, as strongly asymptotically developable functions are bounded in subpolysectors and the -algebra is stable by derivation.
(2) (3). The subpolysector is 1-regular in the sense of H. Whitney: for every , there exists a neighborhood of , and a constant such that, if , a rectifiable curve exists in joining and and such that its length satisfies a bound
Given , let us take a sequence in converging to . If , , a curve exists joining and , and such that . Then
where is a bound for the first derivatives of . The sequence is a Cauchy sequence, so can be extended to . The same argument allows us to extend to all the derivatives of .
As is 1-regular, these extensions define a -function in the sense of Whitney, and the result follows.
(3) (1). Consider a extension of , and define
on , functions that patch together giving a holomorphic function in . These functions define a total family of coefficients . Taylor integral formula allows us to show that, on ,
where is a bound of on .
In classical asymptotics in one variable, Borel–Ritt theorem is of great utility: It says that every formal power series is the asymptotic expansion of some function in an arbitrarily chosen sector. There is an analogue for strong asymptotic expansions, as follows:
Theorem 1.2 (Borel-Ritt)
Given a consistent family of coefficients on a polysector , there exists a function such that .
Sketch of proof
In [8], Majima proves a weaker result. More precisely, he shows that given a formal power series , there exists a function such that . He follows the same idea as in the classical proof in one variable: from the expansion
he constructs a function
defining and appropriately in order to guarantee that the previous expression defines a holomorphic function in the polysector, and the bounds of the definition of the strong asymptotic expansion are verified. A modification of this proof is presented in [9] to show the general case stated here. He employs induction on , assuming in each step that part of the functions are zero.
Let us mention that another proof may be done as follows: a consistent family verifies regularity conditions (in fact, it is formally holomorphic), and so, defines a -function in the sense of Whitney (see [19] for precise definitions and details). So, there exists , a function, defined in a neighborhood of in , such that the restriction of its derivatives coincide with the functions . Truncated Laplace transform of defines an element of and this allows us to conclude.
Strong asymptotic expansions may be defined from one variable asymptotic expansions using functional analysis techniques. For, let us consider the space . For every and , define
This number exists, by Theorem 1.1, and defines a family of seminorms , that provides with a Frechet space structure. If is a Frechet space and a polysector, the space of strongly asymptotically developable functions with values in may be defined. It turns out that there are natural isomorphisms [17], and this allows us to make recurrence on the number of variables.
Returning to Theorem 1.1, let us denote , that is, the space of holomorphic functions in that are in the sense of Whitney in the compact set . Due to the regularity of , this implies that they can be extended as a -function to the whole space . So, we have that . The space , as a subspace of , is a nuclear space [20, 21], and hence, is also nuclear. As is dense in , it can be shown that in fact, , where denotes the topological tensor product, as defined by A. Grothendieck. Precise details may be found in [22].
Let us comment briefly further properties of strong asymptotic expansions.
1.
Consider a multidirection
on a polysector
;
, where
. Assume that
is a holomorphic and bounded function having a strong asymptotic expansion on
: the bounds are verified when restricting to this multidirection (which in fact defines a
-dimensional real space). Then
, that is, the asymptotic expansion exists in the whole polysector. This result is shown in [23], and generalizes a result for the one variable case stated by Fruchard and Zhang [24].
2.
From the sheaf of Whitney
-functions, Honda and Prelli construct in [25] the sheaf of strongly asymptotically developable functions by applying a functor of specialization. This functional setting appears to be rather interesting for future applications, and it deserves further development.
Most of the main results presented in this section have been stated in the context of the so-called Poincaré asymptotics. In the Gevrey case, they are still valid, with more or less straightforward modifications. In the literature on the subject you can find complete statements and proofs. Let us, nevertheless, mention some interesting issues concerning the Gevrey case.
For, recall that in one variable, Watson's lemma says that if has a series as -Gevrey asymptotic expansion, and the opening of the sector is strictly greater than , then is unique, and therefore is called the sum of in .
In the context of strong asymptotic expansions, a similar result is the following one:
Theorem 1.3
Let be a polysector, having a total family of coefficients as -Gevrey strong asymptotic expansion. Then, if for some , the opening of is greater than ,
