Slice hyperholomorphic Schur analysis

This book defines and examines the counterpart of Schur functions and Schur analysis in the slice hyperholomorphic setting. It is organized into three parts: the first introduces readers to classical Schur analysis, while the second offers background material on quaternions, slice hyperholomorphic functions, and quaternionic functional analysis. The third part represents the core of the book and explores quaternionic Schur analysis and its various applications. The book includes previously unpublished results and provides the basis for new directions of research.

Prologue; Part I Classical Schur Analysis; Chapter 1 Preliminaries; 1.1 Some history; 1.2 Krein spaces, Pontryagin spaces, and negative squares; 1.3 The Wiener algebra; 1.4 The Nehari extension problem; 1.5 The Carathéodory-Toeplitz extension problem; 1.6 Various classes of functions and realization theorems; Chapter 2 Rational Functions; 2.1 Rational functions and minimal realizations; 2.2 Minimal factorization; 2.3 Rational functions J-unitary on the imaginary line; 2.4 Rational functions J-unitary on the unit circle; Chapter 3 Schur Analysis; 3.1 The Schur algorithm.

3.2 Interpolation problems3.3 First-order discrete systems; 3.4 The Schur algorithm and reproducing kernel spaces; Part II Quaternionic Analysis; Chapter 4Finite-dimensional Preliminaries; 4.1 Some preliminaries on quaternions; 4.2 Polynomials with quaternionic coefficients; 4.3 Matrices with quaternionic entries; 4.4 Matrix equations; Chapter 5 Quaternionic Functional Analysis; 5.1 Quaternionic locally convex linear spaces; 5.2 Quaternionic inner product spaces; 5.3 Quaternionic Hilbert spaces. Main properties; 5.4 Partial majorants; 5.5 Majorant topologies and inner product spaces.

5.6 Quaternionic Hilbert spac…