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E-raamat: Slice Hyperholomorphic Schur Analysis

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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 The book includes previously unpublished results and provides the basis for new directions of research.

Part I: Classical Schur analysis.- Preliminaries.- Rational functions.- Schur analysis.- Part II: Quaternionic analysis.- Finite dimensional preliminaries.- Quaternionic functional analysis.- Slice hyperholomorphic functions.- Slice hyperholomorphic operator-valued functions.- Part III: Quaternionic Schur analysis.- Reproducing kernel spaces and realizations.- Rational slice hyperholomorphic functions.- First applications: scalar interpolation and first order discrete systems.- Interpolation: operator-valued case.

Arvustused

Purpose of the book is to define and study the counterpart of Schur functions and Schur analysis in the context of slice hyperholomorphic quaternionic functions. the book is an important contribution into each of the two theoriesthose of Schur analysis and of slice hyperholomorphic functions. (Michael Shapiro, Mathematical Reviews, September, 2017)

Prologue ix
Part I Classical Schur Analysis
1 Preliminaries
1.1 Some history
3(3)
1.2 Krein spaces, Pontryagin spaces, and negative squares
6(4)
1.3 The Wiener algebra
10(2)
1.4 The Nehari extension problem
12(3)
1.5 The Caratheodory--Toeplitz extension problem
15(1)
1.6 Various classes of functions and realization theorems
16(7)
2 Rational Functions
2.1 Rational functions and minimal realizations
23(3)
2.2 Minimal factorization
26(2)
2.3 Rational functions J-unitary on the imaginary line
28(3)
2.4 Rational functions J-unitary on the unit circle
31(6)
3 Schur Analysis
3.1 The Schur algorithm
37(2)
3.2 Interpolation problems
39(2)
3.3 First-order discrete systems
41(4)
3.4 The Schur algorithm and reproducing kernel spaces
45(6)
Part II Quaternionic Analysis
4 Finite-dimensional Preliminaries
4.1 Some preliminaries on quaternions
51(3)
4.2 Polynomials with quaternionic coefficients
54(3)
4.3 Matrices with quaternionic entries
57(8)
4.4 Matrix equations
65(6)
5 Quaternionic Functional Analysis
5.1 Quaternionic locally convex linear spaces
71(3)
5.2 Quaternionic inner product spaces
74(5)
5.3 Quaternionic Hilbert spaces. Main properties
79(5)
5.4 Partial major ants
84(4)
5.5 Majorant topologies and inner product spaces
88(8)
5.6 Quaternionic Hilbert spaces. Weak topology
96(4)
5.7 Quaternionic Pontryagin spaces
100(5)
5.8 Quaternionic Krein spaces
105(4)
5.9 Positive definite functions and reproducing kernel quaternionic Hilbert spaces
109(2)
5.10 Negative squares and reproducing kernel quaternionic Pontryagin spaces
111(6)
6 Slice Hyperholomorphic Functions
6.1 The scalar case
117(20)
6.2 The Hardy space of the unit ball
137(6)
6.3 Blaschke products (unit ball case)
143(7)
6.4 The Wiener algebra
150(7)
6.5 The Hardy space of the open half-space
157(4)
6.6 Blaschke products (half-space case)
161(4)
7 Operator-valued Slice Hyperholomorphic Functions
7.1 Definition and main properties
165(6)
7.2 S-spectrum and S-resolvent operator
171(4)
7.3 Functional calculus
175(6)
7.4 Two results on slice hyperholomorphic extension
181(2)
7.5 Slice hyperholomorphic kernels
183(6)
7.6 The space H2H(B) and slice backward-shift invariant subspaces
189(8)
Part III Quaternionic Schur Analysis
8 Reproducing Kernel Spaces and Realizations
8.1 Classes of functions
197(3)
8.2 The Potapov--Ginzburg transform
200(2)
8.3 Schur and generalized Schur functions of the ball
202(10)
8.4 Contractive multipliers, inner multipliers and Beurling-Lax theorem
212(6)
8.5 A theorem on convergence of Schur multipliers
218(2)
8.6 The structure theorem
220(2)
8.7 Caratheodory and generalized Caratheodory functions
222(5)
8.8 Schur and generalized Schur functions of the half-space
227(8)
8.9 Herglotz and generalized Herglotz functions
235(6)
9 Rational Slice Hyperholomorphic Functions
9.1 Definition and first, properties
241(5)
9.2 Minimal realizations
246(4)
9.3 Realizations of unitary rational functions
250(3)
9.4 Rational slice hyperholomorphic functions
253(6)
9.5 Linear fractional transformation
259(2)
9.6 Backward-shift operators
261(5)
10 First Applications: Scalar Interpolation and First-order Discrete Systems
10.1 The Schur algorithm
266(4)
10.2 A particular case
270(4)
10.3 The reproducing kernel method
274(1)
10.4 Caratheodory--Fejer interpolation
275(7)
10.5 Boundary interpolation
282(13)
10.6 First-order discrete linear systems
295(3)
10.7 Discrete systems: the rational case
298(11)
11 Interpolation: Operator-valued Case
11.1 Formulation of the interpolation problems
309(3)
11.2 The problem IP(H2H(B)): the non-degenerate case
312(3)
11.3 Left-tangential interpolation in S(H1, H2, B)
315(4)
11.4 Interpolation in S(H1, H2, B). The non-degenerate case
319(5)
11.5 Interpolation: The case of a finite number of interpolating conditions
324(4)
11.6 Leech's theorem
328(3)
11.7 Interpolation in S(H1, H2, B). Nondegenerate case: Sufficiency
331(2)
Epilogue 333(2)
Bibliography 335(20)
Index 355(6)
Notation Index 361
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