Daniel Alpay Books

This book provides the foundations for a rigorous theory of functional analysis with bicomplex scalars. It begins with a detailed study of bicomplex and hyperbolic numbers and then defines the notion of bicomplex modules. After introducing a number of norms and inner products on such modules (some of which appear in this volume for the first time), the authors develop the theory of linear functionals and linear operators on bicomplex modules. All of this may serve for many different developments, just like the usual functional analysis with complex scalars and in this book it serves as the foundational material for the construction and study of a bicomplex version of the well known Schur analysis.

Focusing on mathematical exercises, this collection guides readers through statistical physics, equilibrium thermodynamics, and information theory, highlighting their interconnections. It covers essential tools from linear algebra, functional analysis, and probability theory, applying them to topics like entropy, machine learning, and quantum channels. The text includes exercises from various domains, with notes providing motivation and hints or solutions for many. Aimed at senior undergraduates and beginning graduate students in mathematics, physics, or engineering, it prepares them for advanced studies.

Focusing on the connections between functional analysis and single-variable function theory, this exercises book for graduate students emphasizes positive definite kernels and reproducing kernel Hilbert spaces. It surveys essential facts from functional analysis and topological vector spaces, followed by an exploration of various Hilbert spaces of analytic functions. This approach provides a comprehensive understanding of the interplay between these mathematical concepts.

Focusing on quaternionic linear operators, this work explores their unique properties compared to complex operators, particularly in the noncommutative realm. It introduces concepts like the S-spectrum and S-resolvent operators while examining de Branges spaces, which serve as quaternionic analogs of analytic function spaces. The study emphasizes specific reproducing kernels in the unit ball or half-space of quaternions, delving into Hilbert, Pontryagin, and Krein spaces and their connections to operator models.

" The volume is dedicated to Lev Sakhnovich, who made fundamental contributions in operator theory and related topics. Besides bibliographic material, it includes a number of selected papers related to Lev Sakhnovich's research interests. The papers are related to operator identities, moment problems, random matrices and linear stochastic systems. " -- The last page of cover

Focusing on the counterpart of Schur functions in the slice hyperholomorphic setting, the book is structured in three parts. The first part covers classical Schur analysis, while the second delves into quaternions and slice hyperholomorphic functions. The core section presents quaternionic Schur analysis, highlighting its applications and previously unpublished results, paving the way for future research directions. This comprehensive examination bridges classical analysis with modern quaternionic techniques, making it a valuable resource for advanced studies.

The origins of Schur analysis lie in a 1917 article by Issai Schur in which he constructed a numerical sequence to correspond to a holomorphic contractive function on the unit disk. These sequences are now known as Schur parameter sequences. Schur analysis has grown significantly since its beginnings in the early twentieth century and now encompasses a wide variety of problems related to several classes of holomorphic functions and their matricial generalizations. These problems include interpolation and moment problems as well as Schur parametrization of particular classes of contractive or nonnegative Hermitian block matrices. This book is primarily devoted to topics related to matrix versions of classical interpolation and moment problems. The major themes include Schur analysis of nonnegative Hermitian block Hankel matrices and the construction of Schur-type algorithms. This book also covers a number of recent developments in orthogonal rational matrix functions, matrix-valued Carathéodory functions and maximal weight solutions for particular matricial moment problems on the unit circle.​

The notions of positive functions and of reproducing kernel Hilbert spaces play an important role in various fields of mathematics, such as stochastic processes, linear systems theory, operator theory, and the theory of analytic functions. Also they are relevant for many applications, for example to statistical learning theory and pattern recognition. The present volume contains a selection of papers which deal with different aspects of reproducing kernel Hilbert spaces. Topics considered include one complex variable theory, differential operators, the theory of self-similar systems, several complex variables, and the non-commutative case. The book is of interest to a wide audience of pure and applied mathematicians, electrical engineers and theoretical physicists.