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Algebraic geometry --- Functional analysis --- Mathematical analysis --- Mathematics --- complexe veranderlijken --- reeksen (wiskunde) --- functies (wiskunde) --- wiskunde

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This book surveys the foundations of the theory of slice regular functions over the quaternions, introduced in 2006, and gives an overview of its generalizations and applications. As in the case of other interesting quaternionic function theories, the original motivations were the richness of the theory of holomorphic functions of one complex variable and the fact that quaternions form the only associative real division algebra with a finite dimension n>2. (Slice) regular functions quickly showed particularly appealing features and developed into a full-fledged theory, while finding applications to outstanding problems from other areas of mathematics. For instance, this class of functions includes polynomials and power series. The nature of the zero sets of regular functions is particularly interesting and strictly linked to an articulate algebraic structure, which allows several types of series expansion and the study of singularities. Integral representation formulas enrich the theory and are fundamental to the construction of a noncommutative functional calculus. Regular functions have a particularly nice differential topology and are useful tools for the construction and classification of quaternionic orthogonal complex structures, where they compensate for the scarcity of conformal maps in dimension four. This second, expanded edition additionally covers a new branch of the theory: the study of regular functions whose domains are not axially symmetric. The volume is intended for graduate students and researchers in complex or hypercomplex analysis and geometry, function theory, and functional analysis in general. From the reviews of the 1st edition: "[The authors] document their own very recent theory of quaternionic regular functions, a development that parallels familiar complex function theory spectacularly well. This user-friendly primary source confirms that quaternionic calculus is not a dead end, and clearly answers a popular question regarding the analogy of complex function theory (complex analysis) with quarternionic variables, making it an excellent basis for a capstone course. Summing Up: Highly recommended. Upper-division undergraduates through professionals." (D. V. Feldman, Choice, Vol. 51 (1), September, 2013)".

Algebraic geometry --- Functional analysis --- Mathematical analysis --- Mathematics --- complexe veranderlijken --- reeksen (wiskunde) --- functies (wiskunde) --- wiskunde

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This book discusses, develops and applies the theory of Vilenkin-Fourier series connected to modern harmonic analysis. The classical theory of Fourier series deals with decomposition of a function into sinusoidal waves. Unlike these continuous waves the Vilenkin (Walsh) functions are rectangular waves. Such waves have already been used frequently in the theory of signal transmission, multiplexing, filtering, image enhancement, code theory, digital signal processing and pattern recognition. The development of the theory of Vilenkin-Fourier series has been strongly influenced by the classical theory of trigonometric series. Because of this it is inevitable to compare results of Vilenkin-Fourier series to those on trigonometric series. There are many similarities between these theories, but there exist differences also. Much of these can be explained by modern abstract harmonic analysis, which studies orthonormal systems from the point of view of the structure of a topological group. The first part of the book can be used as an introduction to the subject, and the following chapters summarize the most recent research in this fascinating area and can be read independently. Each chapter concludes with historical remarks and open questions. The book will appeal to researchers working in Fourier and more broad harmonic analysis and will inspire them for their own and their students' research. Moreover, researchers in applied fields will appreciate it as a sourcebook far beyond the traditional mathematical domains.

Harmonic analysis. Fourier analysis --- Mathematical analysis --- Mathematics --- Fourieranalyse --- analyse (wiskunde) --- reeksen (wiskunde) --- Fourierreeksen --- mathematische modellen --- wiskunde

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This textbook covers the majority of traditional topics of infinite sequences and series, starting from the very beginning - the definition and elementary properties of sequences of numbers, and ending with advanced results of uniform convergence and power series. The text is aimed at university students specializing in mathematics and natural sciences, and at all the readers interested in infinite sequences and series. It is designed for the reader who has a good working knowledge of calculus. No additional prior knowledge is required. The text is divided into five chapters, which can be grouped into two parts: the first two chapters are concerned with the sequences and series of numbers, while the remaining three chapters are devoted to the sequences and series of functions, including the power series. Within each major topic, the exposition is inductive and starts with rather simple definitions and/or examples, becoming more compressed and sophisticated as the course progresses. Each key notion and result is illustrated with examples explained in detail. Some more complicated topics and results are marked as complements and can be omitted on a first reading. The text includes a large number of problems and exercises, making it suitable for both classroom use and self-study. Many standard exercises are included in each section to develop basic techniques and test the understanding of key concepts. Other problems are more theoretically oriented and illustrate more intricate points of the theory, or provide counterexamples to false propositions which seem to be natural at first glance. Solutions to additional problems proposed at the end of each chapter are provided as an electronic supplement to this book.

Functional analysis --- Differential equations --- Mathematical analysis --- Mathematics --- analyse (wiskunde) --- reeksen (wiskunde) --- mathematische modellen --- wiskunde

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Algebraic topology --- Functional analysis --- Harmonic analysis. Fourier analysis --- Differential equations --- Mathematical analysis --- Mathematics --- Fourieranalyse --- analyse (wiskunde) --- topologie (wiskunde) --- reeksen (wiskunde) --- mathematische modellen --- wiskunde