The mathematical theory for many application areas depends on a deep understanding of the theory of moments. These areas include medical imaging, signal processing, computer visualization, and data science. The problem of moments has also found novel applications to areas such as control theory, image analysis, signal processing, polynomial optimization, and statistical big data. The Classical Moment Problem and Some Related Questions in Analysis presents: a unified treatment of the development of the classical moment problem from the late 19th century to the middle of the 20th century, important connections between the moment problem and many branches of analysis, a unified exposition of important classical results, which are difficult to read in the original journals, and a strong foundation for many areas in modern applied mathematics.

This concise, well-written handbook provides a distillation of real variable theory with a particular focus on the subject's significant applications to differential equations and Fourier analysis. Ample examples and brief explanations---with very few proofs and little axiomatic machinery---are used to highlight all the major results of real analysis, from the basics of sequences and series to the more advanced concepts of Taylor and Fourier series, Baire Category, and the Weierstrass Approximation Theorem. Replete with realistic, meaningful applications to differential equations, boundary value problems, and Fourier analysis, this unique work is a practical, hands-on manual of real analysis that is ideal for physicists, engineers, economists, and others who wish to use the fruits of real analysis but who do not necessarily have the time to appreciate all of the theory. Valuable as a comprehensive reference, a study guide for students, or a quick review, "A Handbook of Real Variables" will benefit a wide audience.

Advanced Mathematics for Engineering Students: The Essential Toolbox provides a concise treatment for applied mathematics. Derived from two semester advanced mathematics courses at the author's university, the book delivers the mathematical foundation needed in an engineering program of study. Other treatments typically provide a thorough but somewhat complicated presentation where students do not appreciate the application. This book focuses on the development of tools to solve most types of mathematical problems that arise in engineering - a "toolbox" for the engineer. It provides an important foundation but goes one step further and demonstrates the practical use of new technology for applied analysis with commercial software packages (e.g., algebraic, numerical and statistical).

This is the second of two volumes containing peer-reviewed research and survey papers based on invited talks at the International Conference on Modern Analysis and Applications. The conference, which was dedicated to the 100th anniversary ofthebirthofMarkKrein,oneofthegreatestmathematiciansofthe20thcentury, was held in Odessa, Ukraine, on April 9-14, 2007. The conference focused on the main ideas, methods, results, and achievements of M.G. Krein. This second volume is devoted to the theory of di?erential operators and mechanics. It opens with the description of the conference and a number of survey papers about the work of M.G. Krein. The main part of the book consists of original research papers presenting the state of the art in the area of di?erential operators. The ?rst volume of these proceedings, entitled Operator Theory and Related Topics, concerns other aspects of the conference. The two volumes will be of - terest to a wide-rangeof readership in pure and applied mathematics, physics and engineering sciences. OperatorTheory: AdvancesandApplications,Vol.191, xi-xv c 2009Birkh. auserVerlagBasel/Switzerland The World Dimension of the Heritage of a Ukrainian Mathematician International Conference "Modern Analysis and Applications" (MAA - 2007) (April 9-14, 2007, Odessa) Yu. BerezanskyandV.Gorbachuk This forum has been dedicated to the centennial birthday anniversary of one of the most prominent mathematicians of the twentieth century Mark Gr- orievich Krein, a corresponding member of the Academy of Sciences of the Ukr. SSR (1907-1989).

Learn the basics of white noise theory with White Noise Distribution Theory. This book covers the mathematical foundation and key applications of white noise theory without requiring advanced knowledge in this area. This instructive text specifically focuses on relevant application topics such as integral kernel operators, Fourier transforms, Laplacian operators, white noise integration, Feynman integrals, and positive generalized functions. Extremely well-written by one of the field's leading researchers, White Noise Distribution Theory is destined to become the definitive introductory resource on this challenging topic.

Presents a systematic study of the common zeros of polynomials in several variables which are related to higher dimensional quadrature. The author uses a new approach which is based on the recent development of orthogonal polynomials in several variables and differs significantly from the previous ones based on algebraic ideal theory. Featuring a great deal of new work, new theorems and, in many cases, new proofs, this self-contained work will be of great interest to researchers in numerical analysis, the theory of orthogonal polynomials and related subjects.

This is the ?rst of two volumes containing peer-reviewed research and survey papers based on invited talks at the International Conference on Modern Analysis and Applications. The conference, which was dedicated to the 100th anniversary ofthebirthofMarkKrein,oneofthegreatestmathematiciansofthe20thcentury, was held in Odessa, Ukraine, on April 9-14, 2007. The conference focused on the main ideas, methods, results, and achievements of M. G. Krein. This?rstvolumeisdevotedtotheoperatortheoryandrelatedtopics. Itopens withthebiographypapersaboutM. G. Kreinandanumberofsurveypapersabout his work. The mainpartof the book consistsof originalresearchpaperspresenting the state of the art in operator theory and its application. The second volume of these proceedings, entitled Di?erential Operators and Mechanics, concerns other aspects of the conference. The two volumes will be of interest to a wide-range of readership in pure and applied mathematics, physics and engineering sciences. The editors are sincerely grateful to the persons who contributed to the preparation of these proceedings: Sergei Marchenko, Myroslav Sushko, Kostyantyn Yusenko and Vladimir Zavalnyuk. Mark Grigorievich Krein, 1907-1989 Operator Theory: Advances and Applications, Vol. 190, xi-xx c 2009 Birkh. auser Verlag Basel/Switzerland Mark Grigorievich Krein (on his 100th birthday anniversary) V. M. Adamyan, D. Z. Arov, Yu. M. Berezansky, V. I. Gorbachuk, M. L. Gorbachuk, V. A. Mikhailets and A. M. Samoilenko April 3, 2007, is the l00th anniversary of the birth of Mark Grigorievich Krein, one of the most celebrated mathematicians of the 20th century, whose whole life was closely connected with Ukraine.

This is a systematic exposition of the basic part of the theory of mea sure and integration. The book is intended to be a usable text for students with no previous knowledge of measure theory or Lebesgue integration, but it is also intended to include the results most com monly used in functional analysis. Our two intentions are some what conflicting, and we have attempted a resolution as follows. The main body of the text requires only a first course in analysis as background. It is a study of abstract measures and integrals, and comprises a reasonably complete account of Borel measures and in tegration for R Each chapter is generally followed by one or more supplements. These, comprising over a third of the book, require some what more mathematical background and maturity than the body of the text (in particular, some knowledge of general topology is assumed) and the presentation is a little more brisk and informal. The material presented includes the theory of Borel measures and integration for n, the general theory of integration for locally compact Hausdorff spaces, and the first dozen results about invariant measures for groups. Most of the results expounded here are conventional in general character, if not in detail, but the methods are less so. The following brief overview may clarify this assertion."

Coupled with its sequel, this book gives a connected, unified exposition of Approximation Theory for functions of one real variable. It describes spaces of functions such as Sobolev, Lipschitz, Besov rearrangement-invariant function spaces and interpolation of operators. Other topics include Weierstrauss and best approximation theorems, properties of polynomials and splines. It contains history and proofs with an emphasis on principal results.

This "Selecta" contains approximately two thirds of the papers my father wrote from 1932 to 1994. These papers are divided into four fields. The first volume contains the papers on 1) Summability and Number Theory and 2) Interpolation. The second volume contains the fields 3) Real and Functional Analysis and 4) Approximation Theory. Each of these four groups of papers is introduced by a review of the contents and significance, respectively of the impact of these papers. The first volume contains, in addition, an autobiography, a complete list of publications, a list of doctoral students and four unpublished essays on mathematics in general: a) A report on the University of Leningrad b) On the work of the mathematical mind c) Proofs in Mathematics d) About Mathematical books. The report on the University of Leningrad, written in the late '40's, is a unique historical document which is still of current interest for several reasons. It is of interest for professional reasons since it contains a com plete description of a mathematics majors' curriculum through his entire course of studies. From it one can see both the changes and invariants of course material as well as the students' course load. Then one can also see the consequences of admittedly extreme political intervention in uni versity affairs. Today we use the term "politically correct," but in those times being politically correct was a matter of life and death. Finally, this is a tragedy of human beings caught in the siege of Leningrad."

The works of George G. Lorentz, spanning more than 60 years, have played a significant role in the development and evolution of mathematical analysis. The papers presented in this volume represent a selection of his best works, along with commentary from his students and colleagues.

Real Analysis is indispensable for in-depth understanding and effective application of methods of modern analysis. This concise and friendly book is written for early graduate students of mathematics or of related disciplines hoping to learn the basics of Real Analysis with reasonable ease. The essential role of Real Analysis in the construction of basic function spaces necessary for the application of Functional Analysis in many fields of scientific disciplines is demonstrated with due explanations and illuminating examples. After the introductory chapter, a compact but precise treatment of general measure and integration is taken up so that readers have an overall view of the simple structure of the general theory before delving into special measures. The universality of the method of outer measure in the construction of measures is emphasized because it provides a unified way of looking for useful regularity properties of measures. The chapter on functions of real variables sits at the core of the book; it treats in detail properties of functions that are not only basic for understanding the general feature of functions but also relevant for the study of those function spaces which are important when application of functional analytical methods is in question. This is then followed naturally by an introductory chapter on basic principles of Functional Analysis which reveals, together with the last two chapters on the space of p-integrable functions and Fourier integral, the intimate interplay between Functional Analysis and Real Analysis. Applications of many of the topics discussed are included to motivate the readers for further related studies; these contain explorations towards probability theory and partial differential equations.

This volume is part of the collaboration agreement between Springer and the ISAAC society. This is the first in the two-volume series originating from the 2020 activities within the international scientific conference "Modern Methods, Problems and Applications of Operator Theory and Harmonic Analysis" (OTHA), Southern Federal University in Rostov-on-Don, Russia. This volume is focused on general harmonic analysis and its numerous applications. The two volumes cover new trends and advances in several very important fields of mathematics, developed intensively over the last decade. The relevance of this topic is related to the study of complex multiparameter objects required when considering operators and objects with variable parameters.

The book has been made more illustrative and self-contained so as to cater to the need of students and teachers at graduate and postgraduate level. It is also meant for engineering students and other professionals as well as competitive examinations. To reinforce and solidify the understanding, some of the chapters have been rearranged and several new exercises and solved examples have been incorporated. The section on limits inferior and superior of sequences is introduced and discussed in detail. Every care has been taken to explain and elucidate the different concepts so as to provide conceptual clarity to the readers.

The book offers an initiation into mathematical reasoning, and into the mathematician's mind-set and reflexes. Specifically, the fundamental operations of calculus--differentiation and integration of functions and the summation of infinite series--are built, with logical continuity (i.e., "rigor"), starting from the real number system. The first chapter sets down precise axioms for the real number system, from which all else is derived using the logical tools summarized in an Appendix. The discussion of the "fundamental theorem of calculus," the focal point of the book, especially thorough. The concluding chapter establishes a significant beachhead in the theory of the Lebesgue integral by elementary means.

This book provides a broad, interdisciplinary overview of non-Archimedean analysis and its applications. Featuring new techniques developed by leading experts in the field, it highlights the relevance and depth of this important area of mathematics, in particular its expanding reach into the physical, biological, social, and computational sciences as well as engineering and technology. In the last forty years the connections between non-Archimedean mathematics and disciplines such as physics, biology, economics and engineering, have received considerable attention. Ultrametric spaces appear naturally in models where hierarchy plays a central role - a phenomenon known as ultrametricity. In the 80s, the idea of using ultrametric spaces to describe the states of complex systems, with a natural hierarchical structure, emerged in the works of Fraunfelder, Parisi, Stein and others. A central paradigm in the physics of certain complex systems - for instance, proteins - asserts that the dynamics of such a system can be modeled as a random walk on the energy landscape of the system. To construct mathematical models, the energy landscape is approximated by an ultrametric space (a finite rooted tree), and then the dynamics of the system is modeled as a random walk on the leaves of a finite tree. In the same decade, Volovich proposed using ultrametric spaces in physical models dealing with very short distances. This conjecture has led to a large body of research in quantum field theory and string theory. In economics, the non-Archimedean utility theory uses probability measures with values in ordered non-Archimedean fields. Ultrametric spaces are also vital in classification and clustering techniques. Currently, researchers are actively investigating the following areas: p-adic dynamical systems, p-adic techniques in cryptography, p-adic reaction-diffusion equations and biological models, p-adic models in geophysics, stochastic processes in ultrametric spaces, applications of ultrametric spaces in data processing, and more. This contributed volume gathers the latest theoretical developments as well as state-of-the art applications of non-Archimedean analysis. It covers non-Archimedean and non-commutative geometry, renormalization, p-adic quantum field theory and p-adic quantum mechanics, as well as p-adic string theory and p-adic dynamics. Further topics include ultrametric bioinformation, cryptography and bioinformatics in p-adic settings, non-Archimedean spacetime, gravity and cosmology, p-adic methods in spin glasses, and non-Archimedean analysis of mental spaces. By doing so, it highlights new avenues of research in the mathematical sciences, biosciences and computational sciences.

This contributed volume showcases research and survey papers devoted to a broad range of topics on functional equations, ordinary differential equations, partial differential equations, stochastic differential equations, optimization theory, network games, generalized Nash equilibria, critical point theory, calculus of variations, nonlinear functional analysis, convex analysis, variational inequalities, topology, global differential geometry, curvature flows, perturbation theory, numerical analysis, mathematical finance and a variety of applications in interdisciplinary topics. Chapters in this volume investigate compound superquadratic functions, the Hyers-Ulam Stability of functional equations, edge degenerate pseudo-hyperbolic equations, Kirchhoff wave equation, BMO norms of operators on differential forms, equilibrium points of the perturbed R3BP, complex zeros of solutions to second order differential equations, a higher-order Ginzburg-Landau-type equation, multi-symplectic numerical schemes for differential equations, the Erdos-Renyi network model, strongly m-convex functions, higher order strongly generalized convex functions, factorization and solution of second order differential equations, generalized topologically open sets in relator spaces, graphical mean curvature flow, critical point theory in infinite dimensional spaces using the Leray-Schauder index, non-radial solutions of a supercritical equation in expanding domains, the semi-discrete method for the approximation of the solution of stochastic differential equations, homotopic metric-interval L-contractions in gauge spaces, Rhoades contractions theory, network centrality measures, the Radon transform in three space dimensions via plane integration and applications in positron emission tomography boundary perturbations on medical monitoring and imaging techniques, the KdV-B equation and biomedical applications.

Banach spaces and algebras are a key topic of pure mathematics. Graham Allan's careful and detailed introductory account will prove essential reading for anyone wishing to specialise in functional analysis and is aimed at final year undergraduates or masters level students. Based on the author's lectures to fourth year students at Cambridge University, the book assumes knowledge typical of first degrees in mathematics, including metric spaces, analytic topology, and complex analysis. However, readers are not expected to be familiar with the Lebesgue theory of measure and integration.
The text begins by giving the basic theory of Banach spaces, including dual spaces and bounded linear operators. It establishes forms of the theorems that are the pillars of functional analysis, including the Banach-Alaoglu, Hahn-Banach, uniform boundedness, open mapping, and closed graph theorems. There are applications to Fourier series and operators on Hilbert spaces.
The main body of the text is an introduction to the theory of Banach algebras. A particular feature is the detailed account of the holomorphic functional calculus in one and several variables; all necessary background theory in one and several complex variables is fully explained, with many examples and applications considered. Throughout, exercises at sections ends help readers test their understanding, while extensive notes point to more advanced topics and sources.
The book was edited for publication by Professor H. G. Dales of Leeds University, following the death of the author in August, 2007.

Current research and applications in nonlinear analysis influenced by Haim Brezis and Louis Nirenberg are presented in this book by leading mathematicians. Each contribution aims to broaden reader's understanding of theories, methods, and techniques utilized to solve significant problems. Topics include: Sobolev Spaces Maximal monotone operators A theorem of Brezis-Nirenberg Operator-norm convergence of the Trotter product formula Elliptic operators with infinitely many variables Pseudo-and quasiconvexities for nonsmooth function Anisotropic surface measures Eulerian and Lagrangian variables Multiple periodic solutions of Lagrangian systems Porous medium equation Nondiscrete Lassonde-Revalski principle Graduate students and researchers in mathematics, physics, engineering, and economics will find this book a useful reference for new techniques and research areas. Haim Brezis and Louis Nirenberg's fundamental research in nonlinear functional analysis and nonlinear partial differential equations along with their years of teaching and training students have had a notable impact in the field.

For courses in undergraduate Analysis and Transition to Advanced Mathematics. Analysis with an Introduction to Proof, Fifth Edition helps fill in the groundwork students need to succeed in real analysis-often considered the most difficult course in the undergraduate curriculum. By introducing logic and emphasizing the structure and nature of the arguments used, this text helps students move carefully from computationally oriented courses to abstract mathematics with its emphasis on proofs. Clear expositions and examples, helpful practice problems, numerous drawings, and selected hints/answers make this text readable, student-oriented, and teacher- friendly.

One of the ways in which topology has influenced other branches of mathematics in the past few decades is by putting the study of continuity and convergence into a general setting. This new edition of Wilson Sutherland's classic text introduces metric and topological spaces by describing some of that influence. The aim is to move gradually from familiar real analysis to abstract topological spaces, using metric spaces as a bridge between the two. The language of metric and topological spaces is established with continuity as the motivating concept. Several concepts are introduced, first in metric spaces and then repeated for topological spaces, to help convey familiarity. The discussion develops to cover connectedness, compactness and completeness, a trio widely used in the rest of mathematics.
Topology also has a more geometric aspect which is familiar in popular expositions of the subject as rubber-sheet geometry', with pictures of Mobius bands, doughnuts, Klein bottles and the like; this geometric aspect is illustrated by describing some standard surfaces, and it is shown how all this fits into the same story as the more analytic developments.
The book is primarily aimed at second- or third-year mathematics students. There are numerous exercises, many of the more challenging ones accompanied by hints, as well as a companion website, with further explanations and examples as well as material supplementary to that in the book."

This "Select a" contains approximately two thirds of the papers my 1932 to 1994. These papers are divided into four fields. father wrote from The first volume contains the papers on 1) Summability and Number Theory and 2) Interpolation. The second volume contains the fields 3) Real and Functional Analysis and 4) Approximation Theory. Each of these four groups of papers is introduced by a review of the contents and significance, respectively of the impact of these papers. The first volume contains, in addition, an autobiography, a complete list of publications, a list of doctoral students and four unpublished essays on mathematics in general: a) A report on the University of Leningrad b) On the work of the mathematical mind c) Proofs in Mathematics d) About Mathematical books. The report on the University of Leningrad, written in the late '40's, is a unique historical document which is still of current interest for several reasons. It is of interest for professional reasons since it contains a com plete description of a mathematics majors' curriculum through his entire course of studies. From it one can see both the changes and invariants of course material as well as the students' course load. Then one can also see the consequences of admittedly extreme political intervention in uni versity affairs. Today we use the term "politically correct," but in those times being politically correct was a matter of life and death."

The rapid development of set theory in the last fifty years, mainly by obtaining plenty of independence results, strongly influenced an understanding of the structure of the real line. This book is devoted to the study of the real line and its subsets taking into account the recent results of set theory. Whenever possible the presentation is done without the full axiom of choice. Since the book is intended to be self-contained, all necessary results of set theory, topology, measure theory, and descriptive set theory are revisited with the purpose of eliminating superfluous use of an axiom of choice. The duality of measure and category is studied in a detailed manner. Several statements pertaining to properties of the real line are shown to be undecidable in set theory. The metamathematics behind set theory is shortly explained in the appendix. Each section contains a series of exercises with additional results.

Our knowledge of objects of complex and potential analysis has been enhanced recently by ideas and constructions of theoretical and mathematical physics, such as quantum field theory, nonlinear hydrodynamics, material science. These are some of the themes of this refereed collection of papers, which grew out of the first conference of the European Science Foundation Networking Programme 'Harmonic and Complex Analysis and Applications' held in Norway 2007.

Typically, undergraduates see real analysis as one of the most difficult courses that a mathematics major is required to take. The main reason for this perception is twofold: Students must comprehend new abstract concepts and learn to deal with these concepts on a level of rigor and proof not previously encountered. A key challenge for an instructor of real analysis is to find a way to bridge the gap between a student's preparation and the mathematical skills that are required to be successful in such a course. Real Analysis: With Proof Strategies provides a resolution to the "bridging-the-gap problem." The book not only presents the fundamental theorems of real analysis, but also shows the reader how to compose and produce the proofs of these theorems. The detail, rigor, and proof strategies offered in this textbook will be appreciated by all readers. Features Explicitly shows the reader how to produce and compose the proofs of the basic theorems in real analysis Suitable for junior or senior undergraduates majoring in mathematics.