Sergey Tikhonov - Böcker

Addressing a significant gap in the study of number series, this book presents an in-depth theory of multiple number series and an exhaustive examination of one-dimensional series. It incorporates overlooked yet essential results alongside recent research advancements. Much of the text is based on the authors' original contributions, particularly in the development of relaxed monotonicity concepts, which have become fundamental tools in Fourier and functional analysis. Each chapter concludes with historical context, aiding readers in understanding the theory's evolution. The book is aimed at a wide audience, ranging from undergraduate students to experts in the field. It offers a modern perspective on the theory, along with detailed introductory chapters that make complex concepts accessible for students. The audience will find the novel contributions enriching and inspiring.

Different aspects of harmonic analysis, complex analysis, sampling theory, approximation theory and related topics are covered in this volume.

This book targets graduate students and researchers who want to learn about Lebesgue spaces and solutions to hyperbolic equations. It is divided into two parts.Part 1 provides an introduction to the theory of variable Lebesgue spaces: Banach function spaces like the classical Lebesgue spaces but with the constant exponent replaced by an exponent function. These spaces arise naturally from the study of partial differential equations and variational integrals with non-standard growth conditions. They have applications to electrorheological fluids in physics and to image reconstruction. After an introduction that sketches history and motivation, the authors develop the function space properties of variable Lebesgue spaces; proofs are modeled on the classical theory. Subsequently, the Hardy-Littlewood maximal operator is discussed. In the last chapter, other operators from harmonic analysis are considered, such as convolution operators and singular integrals. The text is mostly self-contained, with only some more technical proofs and background material omitted. Part 2 gives an overview of the asymptotic properties of solutions to hyperbolic equations and systems with time-dependent coefficients. First, an overview of known results is given for general scalar hyperbolic equations of higher order with constant coefficients. Then strongly hyperbolic systems with time-dependent coefficients are considered. A feature of the described approach is that oscillations in coefficients are allowed. Propagators for the Cauchy problems are constructed as oscillatory integrals by working in appropriate time-frequency symbol classes. A number of examples is considered and the sharpness of results is discussed. An exemplary treatment of dissipative terms shows how effective lower order terms can change asymptotic properties and thus complements the exposition.

​This book provides an introduction to h-harmonics and Dunkl transforms. These are extensions of the ordinary spherical harmonics and Fourier transforms, in which the usual Lebesgue measure is replaced by a reflection-invariant weighted measure. The authors’ focus is on the analysis side of both h-harmonics and Dunkl transforms.Graduate students and researchers working in approximation theory, harmonic analysis, and functional analysis will benefit from this book.

This book systematically presents recent fundamental results on greedy approximation with respect to bases.Motivated by numerous applications, the last decade has seen great successes in studying nonlinear sparse approximation. Recent findings have established that greedy-type algorithms are suitable methods of nonlinear approximation in both sparse approximation with respect to bases and sparse approximation with respect to redundant systems. These insights, combined with some previous fundamental results, form the basis for constructing the theory of greedy approximation. Taking into account the theoretical and practical demand for this kind of theory, the book systematically elaborates a theoretical framework for greedy approximation and its applications. The book addresses the needs of researchers working in numerical mathematics, harmonic analysis, and functional analysis. It quickly takes the reader from classical results to the latest frontier, but is written at the level of a graduate course and does not require a broad background in the field.

Different facets of interplay between harmonic analysis andapproximation theory are covered in this volume.  The topics included areFourier analysis, function spaces, optimization theory, partial differentialequations, and their links to modern developments in the approximation theory.The articles of this collection were originated from two events. The first eventtook place during the 9th ISAAC Congress in Krakow, Poland, 5th-9th August2013, at the section “ApproximationTheory and Fourier Analysis”. The secondevent was the conference on Fourier Analysis and Approximation Theory in theCentre de Recerca  Matemàtica (CRM), Barcelona, during 4th-8th November 2013,organized by the editors of this volume. All articles selected to be part ofthis collection were carefully reviewed.

Different facets of interplay between harmonic analysis andapproximation theory are covered in this volume. The topics included areFourier analysis, function spaces, optimization theory, partial differentialequations, and their links to modern developments in the approximation theory.