Slice hyperholomorphic Schur analysis /
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| Author / Creator: | Alpay, Daniel, author. |
|---|---|
| Imprint: | Cham, Switzerland : Birkhäuser, 2016. |
| Description: | 1 online resource |
| Language: | English |
| Series: | Operator theory: Advances and applications, 0255-0156 ; volume 256 Operator theory, advances and applications ; v. 256. |
| Subject: | |
| Format: | E-Resource Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/11270005 |
Table of Contents:
- 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 spaces. Weak topology5.7 Quaternionic Pontryagin spaces; 5.8 Quaternionic Krein spaces; 5.9 Positive definite functions and reproducing kernel quaternionic Hilbert spaces; 5.10 Negative squares and reproducing kernel quaternionic Pontryagin spaces; Chapter 6 Slice Hyperholomorphic Functions; 6.1 The scalar case; 6.2 The Hardy space of the unit ball; 6.3 Blaschke products (unit ball case); 6.4 The Wiener algebra; 6.5 The Hardy space of the open half-space; 6.6 Blaschke products (half-space case); Chapter 7 Operator-valued Slice Hyperholomorphic Functions.
- 7.1 Definition and main properties7.2 S-spectrum and S-resolvent operator; 7.3 Functional calculus; 7.4 Two results on slice hyperholomorphic extension; 7.5 Slice hyperholomorphic kernels; 7.6 The space H2H(B) and slice backward-shift invariant subspaces; Part III Quaternionic Schur Analysis; Chapter 8 Reproducing Kernel Spaces and Realizations; 8.1 Classes of functions; 8.2 The Potapov-Ginzburg transform; 8.3 Schur and generalized Schur functions of the ball; 8.4 Contractive multipliers, inner multipliers and Beurling-Lax theorem; 8.5 A theorem on convergence of Schur multipliers.
- 8.6 The structure theorem8.7 Carathéodory and generalizedCarathéodory functions; 8.8 Schur and generalized Schur functions of the half-space; 8.9 Herglotz and generalized Herglotz functions; Chapter 9 Rational Slice HyperholomorphicFunctions; 9.1 Definition and first properties; 9.2 Minimal realizations; 9.3 Realizations of unitary rational functions; 9.4 Rational slice hyperholomorphic functions; 9.5 Linear fractional transformation; 9.6 Backward-shift operators; Chapter 10 First Applications: Scalar Interpolation and First-order Discrete Systems; 10.1 The Schur algorithm.