Functions with Disconnected Spectrum

Functions with Disconnected Spectrum
Author :
Publisher : American Mathematical Soc.
Total Pages : 152
Release :
ISBN-10 : 9781470428891
ISBN-13 : 147042889X
Rating : 4/5 (91 Downloads)

Synopsis Functions with Disconnected Spectrum by : Alexander M. Olevskii

The classical sampling problem is to reconstruct entire functions with given spectrum S from their values on a discrete set L. From the geometric point of view, the possibility of such reconstruction is equivalent to determining for which sets L the exponential system with frequencies in L forms a frame in the space L2(S). The book also treats the problem of interpolation of discrete functions by analytic ones with spectrum in S and the problem of completeness of discrete translates. The size and arithmetic structure of both the spectrum S and the discrete set L play a crucial role in these problems. After an elementary introduction, the authors give a new presentation of classical results due to Beurling, Kahane, and Landau. The main part of the book focuses on recent progress in the area, such as construction of universal sampling sets, high-dimensional and non-analytic phenomena. The reader will see how methods of harmonic and complex analysis interplay with various important concepts in different areas, such as Minkowski's lattice, Kolmogorov's width, and Meyer's quasicrystals. The book is addressed to graduate students and researchers interested in analysis and its applications. Due to its many exercises, mostly given with hints, the book could be useful for undergraduates.

Function Theory and ℓp Spaces

Function Theory and ℓp Spaces
Author :
Publisher : American Mathematical Soc.
Total Pages : 219
Release :
ISBN-10 : 9781470455934
ISBN-13 : 1470455935
Rating : 4/5 (34 Downloads)

Synopsis Function Theory and ℓp Spaces by : Raymond Cheng

The classical ℓp sequence spaces have been a mainstay in Banach spaces. This book reviews some of the foundational results in this area (the basic inequalities, duality, convexity, geometry) as well as connects them to the function theory (boundary growth conditions, zero sets, extremal functions, multipliers, operator theory) of the associated spaces ℓpA of analytic functions whose Taylor coefficients belong to ℓp. Relations between the Banach space ℓp and its associated function space are uncovered using tools from Banach space geometry, including Birkhoff-James orthogonality and the resulting Pythagorean inequalities. The authors survey the literature on all of this material, including a discussion of the multipliers of ℓpA and a discussion of the Wiener algebra ℓ1A. Except for some basic measure theory, functional analysis, and complex analysis, which the reader is expected to know, the material in this book is self-contained and detailed proofs of nearly all the results are given. Each chapter concludes with some end notes that give proper references, historical background, and avenues for further exploration.

Banach Algebras and the General Theory of *-Algebras: Volume 1, Algebras and Banach Algebras

Banach Algebras and the General Theory of *-Algebras: Volume 1, Algebras and Banach Algebras
Author :
Publisher : Cambridge University Press
Total Pages : 820
Release :
ISBN-10 : 0521366372
ISBN-13 : 9780521366373
Rating : 4/5 (72 Downloads)

Synopsis Banach Algebras and the General Theory of *-Algebras: Volume 1, Algebras and Banach Algebras by : Theodore W. Palmer

This is the first volume of a two volume set that provides a modern account of basic Banach algebra theory including all known results on general Banach *-algebras. This account emphasizes the role of *-algebraic structure and explores the algebraic results that underlie the theory of Banach algebras and *-algebras. The first volume, which contains previously unpublished results, is an independent, self-contained reference on Banach algebra theory. Each topic is treated in the maximum interesting generality within the framework of some class of complex algebras rather than topological algebras. Proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, particularly in noncommutative harmonic analysis, and an extensive bibliography. Volume II is forthcoming.

Functions with Disconnected Spectrum

Functions with Disconnected Spectrum
Author :
Publisher :
Total Pages :
Release :
ISBN-10 : 1470432161
ISBN-13 : 9781470432164
Rating : 4/5 (61 Downloads)

Synopsis Functions with Disconnected Spectrum by : Alexander Ulanovskii

The classical sampling problem is to reconstruct entire functions with given spectrum S from their values on a discrete set L. From the geometric point of view, the possibility of such reconstruction is equivalent to determining for which sets L the exponential system with frequencies in L forms a frame in the space L^2(S). The book also treats the problem of interpolation of discrete functions by analytic ones with spectrum in S and the problem of completeness of discrete translates. The size and arithmetic structure of both the spectrum S and the discrete set L play a crucial role in these p.

Excursions in Harmonic Analysis, Volume 6

Excursions in Harmonic Analysis, Volume 6
Author :
Publisher : Springer Nature
Total Pages : 444
Release :
ISBN-10 : 9783030696375
ISBN-13 : 3030696375
Rating : 4/5 (75 Downloads)

Synopsis Excursions in Harmonic Analysis, Volume 6 by : Matthew Hirn

John J. Benedetto has had a profound influence not only on the direction of harmonic analysis and its applications, but also on the entire community of people involved in the field. The chapters in this volume – compiled on the occasion of his 80th birthday – are written by leading researchers in the field and pay tribute to John’s many significant and lasting achievements. Covering a wide range of topics in harmonic analysis and related areas, these chapters are organized into four main parts: harmonic analysis, wavelets and frames, sampling and signal processing, and compressed sensing and optimization. An introductory chapter also provides a brief overview of John’s life and mathematical career. This volume will be an excellent reference for graduate students, researchers, and professionals in pure and applied mathematics, engineering, and physics.

Sampling: Theory and Applications

Sampling: Theory and Applications
Author :
Publisher : Springer Nature
Total Pages : 210
Release :
ISBN-10 : 9783030362911
ISBN-13 : 3030362914
Rating : 4/5 (11 Downloads)

Synopsis Sampling: Theory and Applications by : Stephen D. Casey

The chapters of this volume are based on talks given at the eleventh international Sampling Theory and Applications conference held in 2015 at American University in Washington, D.C. The papers highlight state-of-the-art advances and trends in sampling theory and related areas of application, such as signal and image processing. Chapters have been written by prominent mathematicians, applied scientists, and engineers with an expertise in sampling theory. Claude Shannon’s 100th birthday is also celebrated, including an introductory essay that highlights Shannon’s profound influence on the field. The topics covered include both theory and applications, such as: • Compressed sensing• Non-uniform and wave sampling• A-to-D conversion• Finite rate of innovation• Time-frequency analysis• Operator theory• Mobile sampling issues Sampling: Theory and Applications is ideal for mathematicians, engineers, and applied scientists working in sampling theory or related areas.

Introduction to Arithmetic Groups

Introduction to Arithmetic Groups
Author :
Publisher : American Mathematical Soc.
Total Pages : 133
Release :
ISBN-10 : 9781470452315
ISBN-13 : 1470452316
Rating : 4/5 (15 Downloads)

Synopsis Introduction to Arithmetic Groups by : Armand Borel

Fifty years after it made the transition from mimeographed lecture notes to a published book, Armand Borel's Introduction aux groupes arithmétiques continues to be very important for the theory of arithmetic groups. In particular, Chapter III of the book remains the standard reference for fundamental results on reduction theory, which is crucial in the study of discrete subgroups of Lie groups and the corresponding homogeneous spaces. The review of the original French version in Mathematical Reviews observes that “the style is concise and the proofs (in later sections) are often demanding of the reader.” To make the translation more approachable, numerous footnotes provide helpful comments.

Combinatorial Convexity

Combinatorial Convexity
Author :
Publisher : American Mathematical Soc.
Total Pages : 148
Release :
ISBN-10 : 9781470467098
ISBN-13 : 1470467097
Rating : 4/5 (98 Downloads)

Synopsis Combinatorial Convexity by : Imre Bárány

This book is about the combinatorial properties of convex sets, families of convex sets in finite dimensional Euclidean spaces, and finite points sets related to convexity. This area is classic, with theorems of Helly, Carathéodory, and Radon that go back more than a hundred years. At the same time, it is a modern and active field of research with recent results like Tverberg's theorem, the colourful versions of Helly and Carathéodory, and the (p,q) (p,q) theorem of Alon and Kleitman. As the title indicates, the topic is convexity and geometry, and is close to discrete mathematics. The questions considered are frequently of a combinatorial nature, and the proofs use ideas from geometry and are often combined with graph and hypergraph theory. The book is intended for students (graduate and undergraduate alike), but postdocs and research mathematicians will also find it useful. It can be used as a textbook with short chapters, each suitable for a one- or two-hour lecture. Not much background is needed: basic linear algebra and elements of (hyper)graph theory as well as some mathematical maturity should suffice.

Generalized Functions, Volume 6

Generalized Functions, Volume 6
Author :
Publisher : American Mathematical Soc.
Total Pages : 450
Release :
ISBN-10 : 9781470426644
ISBN-13 : 1470426641
Rating : 4/5 (44 Downloads)

Synopsis Generalized Functions, Volume 6 by : I. M. Gel′fand

The first systematic theory of generalized functions (also known as distributions) was created in the early 1950s, although some aspects were developed much earlier, most notably in the definition of the Green's function in mathematics and in the work of Paul Dirac on quantum electrodynamics in physics. The six-volume collection, Generalized Functions, written by I. M. Gel′fand and co-authors and published in Russian between 1958 and 1966, gives an introduction to generalized functions and presents various applications to analysis, PDE, stochastic processes, and representation theory. The unifying theme of Volume 6 is the study of representations of the general linear group of order two over various fields and rings of number-theoretic nature, most importantly over local fields (p-adic fields and fields of power series over finite fields) and over the ring of adeles. Representation theory of the latter group naturally leads to the study of automorphic functions and related number-theoretic problems. The book contains a wealth of information about discrete subgroups and automorphic representations, and can be used both as a very good introduction to the subject and as a valuable reference.

Chaos on the Interval

Chaos on the Interval
Author :
Publisher : American Mathematical Soc.
Total Pages : 231
Release :
ISBN-10 : 9781470429560
ISBN-13 : 147042956X
Rating : 4/5 (60 Downloads)

Synopsis Chaos on the Interval by : Sylvie Ruette

The aim of this book is to survey the relations between the various kinds of chaos and related notions for continuous interval maps from a topological point of view. The papers on this topic are numerous and widely scattered in the literature; some of them are little known, difficult to find, or originally published in Russian, Ukrainian, or Chinese. Dynamical systems given by the iteration of a continuous map on an interval have been broadly studied because they are simple but nevertheless exhibit complex behaviors. They also allow numerical simulations, which enabled the discovery of some chaotic phenomena. Moreover, the “most interesting” part of some higher-dimensional systems can be of lower dimension, which allows, in some cases, boiling it down to systems in dimension one. Some of the more recent developments such as distributional chaos, the relation between entropy and Li-Yorke chaos, sequence entropy, and maps with infinitely many branches are presented in book form for the first time. The author gives complete proofs and addresses both graduate students and researchers.