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E-raamat: Bitangential Direct and Inverse Problems for Systems of Integral and Differential Equations

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Bitangential Direct and Inverse Problems for Systems of Integral and Differential EquationsSuurem pilt
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"This largely self-contained treatment surveys, unites and extends some 20 years of research on direct and inverse problems for canonical systems of integral and differential equations and related systems. Five basic inverse problems are studied in whichthe main part of the given data is either a monodromy matrix, an input scattering matrix, an input impedance matrix, a matrix-valued spectral function, or an asymptotic scattering matrix. The corresponding direct problems are also treated. The book incorporates introductions to the theory of matrix-valued entire functions, reproducing kernel Hilbert spaces of vector-valued entire functions (with special attention to two important spaces introduced by L. de Branges), the theory of J-inner matrix-valued functions and their application to bitangential interpolation and extension problems, which can be used independently for courses and seminars in analysis or for self-study. A number of examples are presented to illustrate the theory"--

"This largely self-contained treatment surveys, unites and extends some 20 years of research on direct and inverse problems for canonical systems of integral and differential equations and related systems. Five basic inverse problems are studied in whichthe main part of the given data is either a monodromy matrix; an input scattering matrix; an input impedance matrix; a matrix valued spectral function; or an asymptotic scattering matrix. The corresponding direct problems are also treated. The book incorporates introductions to the theory of matrix valued entire functions, reproducing kernel Hilbert spaces of vector valued entire functions (with special attention to two important spaces introduced by L. de Branges), the theory of J-inner matrix valued functions and their application to bitangential interpolation and extension problems, which can be used independently for courses and seminars in analysis or for self-study. A number of examples are presented to illustrate the theory"--

An essentially self-contained treatment ideal for mathematicians, physicists or engineers whose research is connected with inverse problems.

Arvustused

'The book provides a unified setting for understanding and codifying a number of seemingly disparate areas of analysis appearing not only in the authors' earlier work, but also in numerous articles of other authors scattered throughout the literature.' Joseph A. Ball, Mathematical Reviews

Muu info

An essentially self-contained treatment ideal for mathematicians, physicists or engineers whose research is connected with inverse problems.
Preface xiii
1 Introduction 1(26)
1.1 The matrizant as a chain of entire J-inner mvf's
2(2)
1.2 Monodromy matrices of regular systems
4(1)
1.3 Canonical integral systems
5(1)
1.4 Singular, right regular and right strongly regular matrizants
6(2)
1.5 Input scattering matrices
8(1)
1.6 Chains of associated pairs of the first kind
9(2)
1.7 The bitangential direct input scattering problem
11(1)
1.8 Bitangential inverse monodromy and inverse scattering problems
12(1)
1.9 The generalized Schur interpolation problem
13(1)
1.10 Identifying matrizants as resolvent matrices when J = Jpq
14(1)
1.11 Input impedance matrices and spectral functions
15(2)
1.12 de Branges spaces
17(2)
1.13 Bitangential direct and inverse input impedance and spectral problems
19(2)
1.14 Krein extension problems and Dirac systems
21(2)
1.15 Direct and inverse problems for Dirac-Krein systems
23(2)
1.16 Supplementary notes
25(2)
2 Canonical systems and related differential equations 27(29)
2.1 Canonical integral systems
27(3)
2.2 Connections with canonical differential systems
30(3)
2.3 The matrizant and its properties
33(3)
2.4 Regular case: Monodromy matrix
36(1)
2.5 Multiplicative integral formulas for matrizants and monodromy matrices; Potapov's theorems
37(6)
2.6 The Feller-Krein string equation
43(4)
2.7 Differential systems with potential
47(2)
2.8 Dirac-Krein systems
49(2)
2.9 The Schrodinger equation
51(2)
2.10 Supplementary notes
53(3)
3 Matrix-valued functions in the Nevanlinna class 56(51)
3.1 Preliminaries on the Nevanlinna class Npxq
58(7)
3.2 Linear fractional transformations and Redheffer transformations
65(4)
3.3 The Riesz-Herglotz-Nevanlinna representation
69(5)
3.4 The class E intersection Npxq of entire mvf's in Npxq
74(3)
3.5 The class Πpxq of mvf's in Npxq with pseudocontinuations
77(2)
3.6 Fourier transforms and Paley-Wiener theorems
79(2)
3.7 Entire inner mvf's
81(3)
3.8 J contractive, J-inner and entire J-inner mvf's
84(7)
3.9 Associated pairs of the first kind
91(2)
3.10 Singular and right (and left) regular J-inner mvf's
93(3)
3.11 Linear fractional transformations of Spxq into itself
96(2)
3.12 Linear fractional transformations in Cpxp and from Spxp into Cpxp
98(4)
3.13 Associated pairs of the second kind
102(3)
3.14 Supplementary notes
105(2)
4 Interpolation problems, resolvent matrices and de Branges spaces 107(51)
4.1 The Nehari problem
108(4)
4.2 The generalized Schur interpolation problem
112(6)
4.3 Right and left strongly regular J-inner mvf's
118(1)
4.4 The generalized Carathdodory interpolation problem
119(5)
4.5 Detour on scalar determinate interpolation problems
124(3)
4.6 The reproducing kernel Hilbert space H(U)
127(8)
4.7 de Branges inclusion theorems
135(2)
4.8 A description of H(W) intersection Lm2
137(3)
4.9 The classes UAR(J) and UDR(J) of A-regular and B-regular J-inner mvf's
140(3)
4.10 de Branges matrices E and de Branges spaces B(E)
143(5)
4.11 A coisometry from H(A) onto B(E)
148(1)
4.12 Formulas for resolvent matrices W element of E intersection U°rsR(jpq)
149(2)
4.13 Formulas for resolvent matrices A element of E intersection U°rsR(Jp)
151(4)
4.14 Supplementary notes
155(3)
5 Chains that are matrizants and chains of associated pairs 158(20)
5.1 Continuous chains of entire J-inner mvf's
158(4)
5.2 Chains that are matrizants
162(5)
5.3 Continuity of chains of associated pairs
167(3)
5.4 Type functions for chains
170(7)
5.5 Supplementary notes
177(1)
6 The bitangential direct input scattering problem 178(24)
6.1 The set Sdscat(dM) of input scattering matrices
178(3)
6.2 Parametrization of Sdscat(dM) in terms of Redheffer transforms
181(2)
6.3 Regular canonical integral systems
183(1)
6.4 Limit balls for input scattering matrices
184(5)
6.5 The full rank case
189(3)
6.6 Rank formulas
192(1)
6.7 Regular systems (= full rank) case
193(1)
6.8 The limit point case
193(2)
6.9 The diagonal case
195(1)
6.10 A Weyl-Titchmarsh like characterization for input scattering matrices
195(5)
6.11 Supplementary notes
200(2)
7 Bitangential direct input impedance and spectral problems 202(39)
7.1 Input impedance matrices
202(4)
7.2 Limit balls for input impedance matrices
206(3)
7.3 Formulas for the ranks of semiradii of the limit ball
209(2)
7.4 Bounded mass functions and full rank end points
211(2)
7.5 The limit point case
213(2)
7.6 The Weyl-Titchmarsh characterization of the input impedance
215(4)
7.7 Spectral functions for canonical systems
219(6)
7.8 Parametrization of the set (H(A))psf
225(5)
7.9 Parametrization of the set Σdpsf(dM) for regular canonical integral systems
230(2)
7.10 Pseudospectral and spectral functions for singular systems
232(5)
7.11 Supplementary notes
237(4)
8 Inverse monodromy problems 241(69)
8.1 Some simple illustrative examples
244(3)
8.2 Extremal solutions when J = Im
247(6)
8.3 Solutions for U element of UAR(J) when J is not = to ±Im
253(5)
8.4 Connections with the Livsic model of a Volterra node
258(11)
8.5 Conditions for the uniqueness of normalized Hamiltonians
269(8)
8.6 Solutions with symplectic and/or real matrizants
277(4)
8.7 Entire homogeneous resolvent matrices
281(4)
8.8 Solutions with homogeneous matrizants
285(6)
8.9 Extremal solutions for J is not = to ±Im
291(3)
8.10 The unicellular case for J is not = to ±Im
294(1)
8.11 Solutions with symmetric type
295(4)
8.12 The inverse rnonodromy problem for 2 x 2 differential systems
299(5)
8.13 Examples of 2 x 2 Hamiltonians with constant determinant
304(4)
8.14 Supplementary notes
308(2)
9 Bitangential Krein extension problems 310(45)
9.1 Helical extension problems
311(6)
9.2 Bitangential helical extension problems
317(3)
9.3 The Krein accelerant extension problem
320(8)
9.4 Continuous analogs of the Schur extension problem
328(6)
9.5 A bitangential generalization of the Schur extension problem
334(4)
9.6 The Nehari extension problem for mvf's in Wiener class
338(9)
9.7 Continuous analogs of the Schur extension problem for mvf's in the Wiener class
347(3)
9.8 Bitangential Schur extension problems in the Wiener class
350(4)
9.9 Supplementary notes
354(1)
10 Bitangential inverse input scattering problems 355(17)
10.1 Existence and uniqueness of solutions
356(1)
10.2 Formulas for the solution of the inverse input scattering problem
357(4)
10.3 Input scattering matrices in the Wiener class
361(1)
10.4 Examples with diagonal mvf's bt1 and bt2
362(9)
10.5 Supplementary notes
371(1)
11 Bitangential inverse input impedance and spectral problems 372(37)
11.1 Existence and uniqueness of solutions
373(2)
11.2 Formulas for the solutions
375(3)
11.3 Input impedance matrices in the Wiener class
378(5)
11.4 Examples with diagonal mvf's bt3 and bt4 = Ip
383(10)
11.5 The bitangential inverse spectral problem
393(3)
11.6 An example
396(10)
11.7 Supplementary notes
406(3)
12 Direct and inverse problems for Dirac-Krein systems 409(41)
12.1 Factoring Hamiltonians corresponding to DK-systems
411(5)
12.2 Matrizants of canonical differential systems corresponding to DK-systems
416(7)
12.3 Direct and inverse monodromy problems for DK-systems
423(1)
12.4 Direct and inverse input scattering problems for DK-systems
424(3)
12.5 Direct and inverse input impedance problems for DK-systems
427(4)
12.6 Direct and inverse spectral problems for DK-systems
431(2)
12.7 The Krein algorithms for the inverse input scattering and impedance problems
433(2)
12.8 The left transform TlUl for Ul element of Wl(jpq)
435(3)
12.9 Asymptotic equivalence matrices
438(1)
12.10 Asymptotic scattering matrices (S-matrices)
438(5)
12.11 The inverse asymptotic scattering problem
443(2)
12.12 More on spectral functions of DK-systems
445(2)
12.13 Supplementary notes
447(3)
References 450(15)
Symbol index 465(4)
Index 469
Damir Z. Arov is Professor in the Division of Applied Mathematics and Informatics at the South-Ukrainian Pedagogical University, Odessa. Harry Dym is Professor Emeritus in the Department of Mathematics at the Weizmann Institute of Science, Rehovot, Israel.
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