Functional analysis and applications

565 Pages·2018·3.05 MB·English

by  Siddiqi, Abul Hasan

Industrial and Applied Mathematics
Abul Hasan Siddiqi
Functional
Analysis and
Applications
Industrial and Applied Mathematics
Editor-in-chief
Abul Hasan Siddiqi, Sharda University, Greater Noida, India
Editorial Board
Zafer Aslan, International Centre for Theoretical Physics, Istanbul, Turkey
M. Brokate, Technical University, Munich, Germany
N.K. Gupta, Indian Institute of Technology Delhi, New Delhi, India
AkhtarKhan,CenterforAppliedandComputationalMathematics,Rochester,USA
Rene Lozi, University of Nice Sophia-Antipolis, Nice, France
Pammy Manchanda, Guru Nanak Dev University, Amritsar, India
M. Zuhair Nashed, University of Central Florida, Orlando, USA
Govindan Rangarajan, Indian Institute of Science, Bengaluru, India
K.R. Sreenivasan, Polytechnic School of Engineering, New York, USA
TheIndustrialandAppliedMathematicsseriespublisheshigh-qualityresearch-level
monographs, lecture notes and contributed volumes focusing on areas where
mathematics is used in a fundamental way, such as industrial mathematics,
bio-mathematics, financial mathematics, applied statistics, operations research and
computer science.
More information about this series at http://www.springer.com/series/13577
Abul Hasan Siddiqi
Functional Analysis
and Applications
123
AbulHasan Siddiqi
Schoolof BasicSciences andResearch
Sharda University
Greater Noida, Uttar Pradesh
India
ISSN 2364-6837 ISSN 2364-6845 (electronic)
Industrial andAppliedMathematics
ISBN978-981-10-3724-5 ISBN978-981-10-3725-2 (eBook)
https://doi.org/10.1007/978-981-10-3725-2
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To
My wife Azra
Preface
Functional analysis was invented and developed in the twentieth century. Besides
being an area of independent mathematical interest, it provides many fundamental
notions essential for modeling, analysis, numerical approximation, and computer
simulation processes of real-world problems. As science and technology are
increasingly refined and interconnected, the demand for advanced mathematics
beyond the basic vector algebra and differential and integral calculus has greatly
increased. There is no dispute on the relevance of functional analysis; however,
there have been differences of opinion among experts about the level and
methodology of teaching functional analysis. In the recent past, its applied nature
has been gaining ground.
The main objective of this book is to present all those results of functional
analysis, which have been frequently applied in emerging areas of science and
technology.
Functional analysis provides basic tools and foundation for areas of vital
importance such as optimization, boundary value problems, modeling real-world
phenomena, finite and boundary element methods, variational equations and
inequalities,inverseproblems,andwaveletandGaboranalysis.Wavelets,formally
invented in the mid-eighties, have found significant applications in image pro-
cessing and partial differential equations. Gabor analysis was introduced in 1946,
gaining popularity since the last decade among the signal processing community
and mathematicians.
The book comprises 15 chapters, an appendix, and a comprehensive updated
bibliography. Chapter 1 is devoted to basic results of metric spaces, especially an
importantfixed-pointtheoremcalledtheBanachcontractionmappingtheorem,and
its applications to matrix, integral, and differential equations. Chapter 2 deals with
basic definitions and examples related to Banach spaces and operators defined on
suchspaces.Asufficientnumberofexamplesarepresentedtomaketheideasclear.
Algebras of operators and properties of convex functionals are discussed. Hilbert
space, an infinite-dimensional analogue of Euclidean space offinite dimension, is
introducedanddiscussedindetailinChap.3.Inaddition,importantresultssuchas
projection theorem, Riesz representation theorem, properties of self-adjoint,
vii
viii Preface
positive, normal, and unitary operators, relationship between bounded linear
operator and bounded bilinear form, and Lax–Milgram lemma dealing with the
existence of solutions of abstract variational problems are presented. Applications
and generalizations of the Lax–Milgram lemma are discussed in Chaps. 7 and 8.
Chapter 4 is devoted to the Hahn–Banach theorem, Banach–Alaoglu theorem,
uniform boundedness principle, open mapping, and closed graph theorems along
withtheconceptofweakconvergenceandweaktopologies.Chapter5providesan
extension of finite-dimensional classical calculus to infinite-dimensional spaces,
which is essential to understand and interpret various current developments of
scienceandtechnology.Moreprecisely,derivativesinthesenseofGâteau,Fréchet,
Clarke (subgradient), and Schwartz (distributional derivative) along with Sobolev
spaces are the main themes of this chapter. Fundamental results concerning exis-
tence and uniqueness of solutions and algorithm for finding solutions of opti-
mization problems aredescribed inChap.6. Variational formulation and existence
of solutions of boundary value problems representing physical phenomena are
described in Chap. 7. Galerkin and Ritz approximation methods are also included.
Finiteelementandboundaryelementmethodsareintroducedandseveraltheorems
concerning error estimation and convergence are proved in Chap. 8. Chapter 9 is
devoted to variational inequalities. A comprehensive account of this elegant
mathematical model in terms of operators is given. Apart from existence and
uniqueness of solutions, error estimation and finite element methods for approxi-
mate solutions and parallel algorithms are discussed. The chapter is mainly based
on the work of one of its inventors, J. L. Lions, and his co-workers and research
students. Activities at the Stampacchia School of Mathematics, Erice, Italy, are
providingimpetustoresearchersinthisfield.Chapter10isdevotedtorudimentsof
spectral theory with applications to inverse problems. We present frame and basis
theory in Hilbert spaces in Chap. 11. Chapter 12 deals with wavelets. Broadly,
waveletanalysisisarefinementofFourieranalysisandhasattractedtheattentionof
researchers in mathematics, physics, and engineering alike. Replacement of the
classical Fourier methods, wherever they have been applied, by emerging wavelet
methodshasresultedindrasticimprovements.Inthischapter,adetailedaccountof
thisexcitingtheoryispresented.Chapter13presentsanintroductiontoapplications
ofwaveletmethodstopartialdifferentialequationsandimageprocessing.Theseare
emerging areas of current interest. There is still a wide scope for further research.
Models and algorithms for removal of an unwanted component (noise) of a signal
are discussed in detail. Error estimation of a given image with its wavelet repre-
sentation in the Besov norm is given. Wavelet frames are comparatively a new
addition to wavelet theory. We discuss their basic properties in Chap. 14. Dennis
Gabor, Nobel Laureate of Physics (1971), introduced windowed Fourier analysis,
now called Gabor analysis, in 1946. Fundamental concepts of this analysis with
certain applications are presented in Chap. 15.
In appendix, we present a resume of the results of topology, real analysis,
calculus,andFourieranalysiswhichweoftenuseinthisbook.Chapters9,12,13,
and 15 contain recent results opening up avenues for further work.
Preface ix
The book is self-contained and provides examples, updated references, and
applications in diverse fields. Several problems are thought-provoking, and many
lead to new results and applications. The book is intended to be a textbook for
graduateorseniorundergraduatestudentsinmathematics.Itcouldalsobeusedfor
an advance course in system engineering, electrical engineering, computer engi-
neering,andmanagementsciences.Theproofsoftheoremsandotheritemsmarked
with an asterisk may be omitted for a senior undergraduate course or a course in
other disciplines. Those who are mainly interested in applications of wavelets and
GaborsystemmaystudyChaps.2,3,and11to15.Readersinterestedinvariational
inequalitiesanditsapplicationsmaypursueChaps.3,8,and9.Inbrief,thisbookis
a handy manual of contemporary analytic and numerical methods in
infinite-dimensional spaces, particularly Hilbert spaces.
Ihaveused amajorpart ofthematerial presentedinthebookwhile teachingat
variousuniversitiesoftheworld.Ihavealsoincorporatedinthisbooktheideasthat
emerged after discussion with some senior mathematicians including Prof. M. Z.
Nashed, Central Florida University; Prof. P. L. Butzer, Aachen Technical
University;Prof.JochimZoweandProf.MichaelKovara,ErlangenUniversity;and
Prof. Martin Brokate, Technical University, Munich.
ItakethisopportunitytothankProf.P.Manchanda,Chairperson,Departmentof
Mathematics, Guru Nanak Dev University, Amritsar, India; Prof. Rashmi
Bhardwaj, Chairperson, Non-linear Dynamics Research Lab, Guru Gobind Singh
Indraprastha University, Delhi, India; and Prof. Q. H. Ansari, AMU/KFUPM, for
their valuable suggestions in editing the manuscript. I also express my sincere
thanks to Prof. M. Al-Gebeily, Prof. S. Messaoudi, Prof. K. M. Furati, and
Prof. A. R. Khan for reading carefully different parts of the book.
Greater Noida, India Abul Hasan Siddiqi
Contents
1 Banach Contraction Fixed Point Theorem . . . . . . . . . . . . . . . . . . . 1
1.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Contraction Fixed Point Theorem by Stefan Banach . . . . . . . . 1
1.3 Application of Banach Contraction Mapping Theorem . . . . . . . 7
1.3.1 Application to Matrix Equation. . . . . . . . . . . . . . . . . . 7
1.3.2 Application to Integral Equation . . . . . . . . . . . . . . . . . 9
1.3.3 Existence of Solution of Differential Equation . . . . . . . 12
1.4 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2 Banach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Basic Results of Banach Spaces. . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Examples of Normed and Banach Spaces . . . . . . . . . . 17
2.3 Closed, Denseness, and Separability. . . . . . . . . . . . . . . . . . . . . 20
2.3.1 Introduction to Closed, Dense, and Separable Sets . . . . 20
2.3.2 Riesz Theorem and Construction of a New Banach
Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.3 Dimension of Normed Spaces. . . . . . . . . . . . . . . . . . . 22
2.3.4 Open and Closed Spheres. . . . . . . . . . . . . . . . . . . . . . 23
2.4 Bounded and Unbounded Operators. . . . . . . . . . . . . . . . . . . . . 25
2.4.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . . . 25
2.4.2 Properties of Linear Operators. . . . . . . . . . . . . . . . . . . 33
2.4.3 Unbounded Operators. . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Representation of Bounded and Linear Functionals. . . . . . . . . . 41
2.6 Space of Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.7 Convex Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.7.1 Convex Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.7.2 Affine Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.7.3 Lower Semicontinuous and Upper Semicontinuous
Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
xi