Discrete and continuous boundary problems / F. V. Atkinson.
Series Mathematics in science and engineering ; v. 8Editor: New York : Academic Press, 1964Descripción: xiv, 570 p. ; 24 cmOtra clasificación: 34.30 (39.20)0.1. Difference and Differential Equations [1] 0.2. The Invariance Property [4] 0.3. The Scalar Case [6] 0.4. All-Pass Transfer Functions [8] 0.5. Inverse Problems [11] 0.6. The General Orthogonal Case [13] 0.7. The Three-Term Recurrence Formula [15] 0.8. The 2-by-2 Symplectic Case [21] 1—Boundary Problems for Rational Functions 1.1. Finite Fourier Series [25] 1.2. The Boundary Problem [27] 1.3. Oscillation Properties [29] 1.4. Eigenfunctions and Orthogonality [31] 1.5. The Spectral Function [35] 1.6. The Characteristic Function [37] 1.7. The First Inverse Problem [39] 1.8. The Second Inverse Problem [42] 1.9. Moment Characterization of the Spectral Function [46] 1.10. Solution of a Moment Problem [51] 2—The Infinite Discrete Case 2.1. A Limiting Procedure [55] 2.2. Convergence of the Fundamental Solution [57] 2.3. Convergence of the Spectral Function [60] 2.4. Convergence of the Characteristic Function [62] 2.5. Eigenvalues and Orthogonality [63] 2.6. Orthogonality and Expansion Theorem [67] 2.7. A Continuous Spectrum [70] 2.8. Moment and Interpolation Problem [71] 2.9. A Mixed Boundary Problem [74] 2.10. A Mixed Expansion Problem [76] 2.11. Further Boundary Problems [81] 3— Discrete Linear Problems 3.1. Problems Linear in the Parameter [83] 3.2. Reduction to Canonical Form [85] 3.3. The Real Axis Case [87] 3.4. The Unit Circle Case [89] 3.5. The Real 2-by-2 Case [90] 3.6. The 2-by-2 Unit Circle Case [92] 3.7. The Boundary Problem on the Real Axis [94] 3.8. The Boundary Problem on the Unit Circle [96] 4— Finite Orthogonal Polynomials 4.1. The Recurrence Relation [97] 4.2. Lagrange-Type Identities [98] 4.3. Oscillatory Properties [100] 4.4. Orthogonality [104] 4.5. Spectral and Characteristic Functions [106] 4.6. The First Inverse Spectral Problem [107] 4.7. The Second Inverse Spectral Problem [111] 4.8. Spectral Functions in General [114] 4.9. Some Continuous Spectral Functions [117] 5— Orthogonal Polynomials The Infinite Case 5.1. Limiting Boundary Problems [119] 5.2. Spectral Functions [120] 5.3. Orthogonality and Expansion Theorem [123] 5.4. Nesting Circle Analysis [125] 55, Limiting Spectral Functions [129] 5,6. Solutions of Summable Square [130] 5 7, Eigenvalues in the Limit-Circle Case [132] 5.8. Limit-Circle, Limit-Point Tests [134] 5.9. Moment Problem [136] 5.10. The Dual Expansion Theorem [138] 6— Matrix Methods for Polynomials 6.1. Orthogonal Polynomials as Jacobi Determinants [142] 6.2. Expansion Theorems, Periodic Boundary Conditions [144] 6.3. Another Method for Separation Theorems [145] 6.4. The Green’s Function [148] 6.5. A Reactance Theorem [150] 6.6. Polynomials with Matrix Coefficients [150] 6.7. Oscillatory Properties [152] 6.8. Orthogonality [157] 6.9. Polynomials in Several Variables [160] 6.10. The Multi-Parameter Oscillation Theorem [162] 6.11. Multi-Dimensional Orthogonality [169] 7— Polynomials Orthogonal on the Unit Circle 7.1. The Recurrence Relation [170] 7.2. The Boundary Problem [172] 7.3. Orthogonality [173] 7.4. The Recurrence Formulas Deduced from the Orthogonality [178] 7.5. Uniqueness of the Spectral Function [182] 7.6. The Characteristic Function [184] 7.7. A Further Orthogonality Result [188] 7.8. Asymptotic Behavior [190] 7.9. Polynomials Orthogonal on a Real Segment [196] 7.10. Continuous and Discrete Analogs [199] 8— Sturm-Liouville Theory 8.1. The Differential Equation [202] 8.2. Existence, Uniqueness, and Bounds for Solutions [205] 8.3. The Boundary Problem [207] 8.4. Oscillatory Properties [209] 8.5. An Interpolator Property [217] 8.6. The Eigenfunction Expansion [222] 8.7. Second-Order Equation with Discontinuities [226] 8.8. The Green’s Function [229] 8.9. Convergence of the Eigenfunction Expansion [232] 8.10. Spectral Functions [238] 8.11. Explicit Expansion Theorem [240] 8.12. Expansions over a Half-Axis [243] 8.13. Nesting Circles [247] 9— The General First-Order Differential System 9.1. Formalities [252] 9.2. The Boundary Problem [255] 9.3. Eigenfunctions and Orthogonality [258] 9.4. The Inhomogeneous Problem [262] 9.5. The Characteristic Function [268] 9.6. The Eigenfunction Expansion [273] 9.7. Convergence of the Eigenfunction Expansion [280] 9.8. Nesting Circles [284] 9.9. Expansion of the Basic Interval [289] 9.10. Limit-Circle Theory [292] 9.11. Solutions of Integrable Square [293] 9.12. The Limiting Process a —> — oo, b —> + oo [298] 10—Matrix Oscillation Theory 10.1. Introduction [300] 10.2. The Matrix Sturm-Liouville Equation [303] 10.3. Separation Theorem for Conjugate Points [308] 10.4. Estimates of Oscillation [312] 10.5. Boundary Problems with a Parameter [317] 10.6. A Fourth-Order Scalar Equation [323] 10.7. The First-Order Equation [328] 10.8. Conjugate Point Problems [332] 10.9. First-Order Equation with Parameter [336] 11—From Differential to Integral Equations 11.1. The Sturm-Liouville Case [339] 11.2. Uniqueness and Existence of Solutions [341] 11.3. Wronskian Identities [348] 11.4. Variation of Parameters [350] 11.5. Analytic Dependence on a Parameter [355] 11.6. Eigenvalues and Orthogonality [356] 11.7. Remarks on the Expansion Theorem [358] 11.8. The Generalized First-Order Matrix Differential Equation [359] 11.9. A Special Case [363] 11.10. The Boundary Problem [364] 12—Asymptotic Theory of Some Integral Equations 12.1. Asymptotically Trigonometric Behavior [366] 12.2. The -Function [371] 12.3. A Non-Self-Adjoint Problem [375] 12.4. The Sturm-Liouville Problem [381] 12.5. Asymptotic Properties for the Generalization of y'' + [k2 + g(x)] y = 0 [384] 12.6. Solutions of Integrable Square [391] 12.7. Analytic Aspects of Asymptotic Theory [393] 12.8. Approximations over a Finite Interval [398] 12.9 Approximation to the Eigenfunctions [408] 12.10. Completeness of the Eigenfunctions [411] Appendix I. Some Compactness Principles for Stieltjes Integrals 1.1. Functions of Bounded Variation [416] 1.2. The Riemann-Stieltjes Integral [418] 1.3. A Convergence Theorem [423] 1.4. The Helly-Bray Theorem [425] 1.5. Infinite Interval and Bounded Integrand [426] 1.6. Infinite Interval with Polynomial Integrand [428] 1.7. A Periodic Case [430] 1.8. The Matrix Extension [431] 1.9. The Multi-Dimensional Case [434] Appendix II. Functions of Negative Imaginary Type II.1. Introduction [436] II.2. The Rational Case [437] II.3. Separation Property in the Meromorphic Case [439] Appendix III. Orthogonality of Vectors 111.1. The Finite-Dimensional Case [441] 111.2. The Infinite-Dimensional Case [442] Appendix IV. Some Stability Results for Linear Systems IV.1. A Discrete Case [447] IV.2. The Case of a Differential Equation [449] IV.3. A Second-Order Differential Equation [450] IV.4. The Mixed or Continuous-Discrete Case [452] IV.5, The Extended Gronwall Lemma [455] Appendix V. Eigenvalues of Varying Matrices V.l. Variational Expressions for Eigenvalues [457] V.2. Continuity and Monotonicity of Eigenvalues [459] V.3. A Further Monotonicity Criterion [461] V.4. Varying Unitary Matrices [464] V.5. Continuation of the Eigenvalues [465] V.6. Monotonicity of the Unit Circle [468] Appendix VI. Perturbation of Bases in Hilbert Space VI. 1. The Basic Result [471] VI.2. Continuous Variation of a Basis [473] VI. 3. Another Result [475] NOTATION AND TERMINOLOGY [476] LIST OF BOOKS AND MONOGRAPHS [478]
| Item type | Home library | Call number | Materials specified | Status | Date due | Barcode | Course reserves |
|---|---|---|---|---|---|---|---|
Libros
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Instituto de Matemática, CONICET-UNS | 34 At875 (Browse shelf) | Available | A-3341 |
Bibliografía: p. 478-480. Referencias bibliográficas incluidas en "Notes" (p. 481-535)
0.1. Difference and Differential Equations [1] --
0.2. The Invariance Property [4] --
0.3. The Scalar Case [6] --
0.4. All-Pass Transfer Functions [8] --
0.5. Inverse Problems [11] --
0.6. The General Orthogonal Case [13] --
0.7. The Three-Term Recurrence Formula [15] --
0.8. The 2-by-2 Symplectic Case [21] --
1—Boundary Problems for Rational Functions --
1.1. Finite Fourier Series [25] --
1.2. The Boundary Problem [27] --
1.3. Oscillation Properties [29] --
1.4. Eigenfunctions and Orthogonality [31] --
1.5. The Spectral Function [35] --
1.6. The Characteristic Function [37] --
1.7. The First Inverse Problem [39] --
1.8. The Second Inverse Problem [42] --
1.9. Moment Characterization of the Spectral Function [46] --
1.10. Solution of a Moment Problem [51] --
2—The Infinite Discrete Case --
2.1. A Limiting Procedure [55] --
2.2. Convergence of the Fundamental Solution [57] --
2.3. Convergence of the Spectral Function [60] --
2.4. Convergence of the Characteristic Function [62] --
2.5. Eigenvalues and Orthogonality [63] --
2.6. Orthogonality and Expansion Theorem [67] --
2.7. A Continuous Spectrum [70] --
2.8. Moment and Interpolation Problem [71] --
2.9. A Mixed Boundary Problem [74] --
2.10. A Mixed Expansion Problem [76] --
2.11. Further Boundary Problems [81] --
3— Discrete Linear Problems --
3.1. Problems Linear in the Parameter [83] --
3.2. Reduction to Canonical Form [85] --
3.3. The Real Axis Case [87] --
3.4. The Unit Circle Case [89] --
3.5. The Real 2-by-2 Case [90] --
3.6. The 2-by-2 Unit Circle Case [92] --
3.7. The Boundary Problem on the Real Axis [94] --
3.8. The Boundary Problem on the Unit Circle [96] --
4— Finite Orthogonal Polynomials --
4.1. The Recurrence Relation [97] --
4.2. Lagrange-Type Identities [98] --
4.3. Oscillatory Properties [100] --
4.4. Orthogonality [104] --
4.5. Spectral and Characteristic Functions [106] --
4.6. The First Inverse Spectral Problem [107] --
4.7. The Second Inverse Spectral Problem [111] --
4.8. Spectral Functions in General [114] --
4.9. Some Continuous Spectral Functions [117] --
5— Orthogonal Polynomials The Infinite Case --
5.1. Limiting Boundary Problems [119] --
5.2. Spectral Functions [120] --
5.3. Orthogonality and Expansion Theorem [123] --
5.4. Nesting Circle Analysis [125] --
55, Limiting Spectral Functions [129] --
5,6. Solutions of Summable Square [130] --
5 7, Eigenvalues in the Limit-Circle Case [132] --
5.8. Limit-Circle, Limit-Point Tests [134] --
5.9. Moment Problem [136] --
5.10. The Dual Expansion Theorem [138] --
6— Matrix Methods for Polynomials --
6.1. Orthogonal Polynomials as Jacobi Determinants [142] --
6.2. Expansion Theorems, Periodic Boundary Conditions [144] --
6.3. Another Method for Separation Theorems [145] --
6.4. The Green’s Function [148] --
6.5. A Reactance Theorem [150] --
6.6. Polynomials with Matrix Coefficients [150] --
6.7. Oscillatory Properties [152] --
6.8. Orthogonality [157] --
6.9. Polynomials in Several Variables [160] --
6.10. The Multi-Parameter Oscillation Theorem [162] --
6.11. Multi-Dimensional Orthogonality [169] --
7— Polynomials Orthogonal on the Unit Circle --
7.1. The Recurrence Relation [170] --
7.2. The Boundary Problem [172] --
7.3. Orthogonality [173] --
7.4. The Recurrence Formulas Deduced from the Orthogonality [178] --
7.5. Uniqueness of the Spectral Function [182] --
7.6. The Characteristic Function [184] --
7.7. A Further Orthogonality Result [188] --
7.8. Asymptotic Behavior [190] --
7.9. Polynomials Orthogonal on a Real Segment [196] --
7.10. Continuous and Discrete Analogs [199] --
8— Sturm-Liouville Theory --
8.1. The Differential Equation [202] --
8.2. Existence, Uniqueness, and Bounds for Solutions [205] --
8.3. The Boundary Problem [207] --
8.4. Oscillatory Properties [209] --
8.5. An Interpolator Property [217] --
8.6. The Eigenfunction Expansion [222] --
8.7. Second-Order Equation with Discontinuities [226] --
8.8. The Green’s Function [229] --
8.9. Convergence of the Eigenfunction Expansion [232] --
8.10. Spectral Functions [238] --
8.11. Explicit Expansion Theorem [240] --
8.12. Expansions over a Half-Axis [243] --
8.13. Nesting Circles [247] --
9— The General First-Order Differential System --
9.1. Formalities [252] --
9.2. The Boundary Problem [255] --
9.3. Eigenfunctions and Orthogonality [258] --
9.4. The Inhomogeneous Problem [262] --
9.5. The Characteristic Function [268] --
9.6. The Eigenfunction Expansion [273] --
9.7. Convergence of the Eigenfunction Expansion [280] --
9.8. Nesting Circles [284] --
9.9. Expansion of the Basic Interval [289] --
9.10. Limit-Circle Theory [292] --
9.11. Solutions of Integrable Square [293] --
9.12. The Limiting Process a —> — oo, b —> + oo [298] --
10—Matrix Oscillation Theory --
10.1. Introduction [300] --
10.2. The Matrix Sturm-Liouville Equation [303] --
10.3. Separation Theorem for Conjugate Points [308] --
10.4. Estimates of Oscillation [312] --
10.5. Boundary Problems with a Parameter [317] --
10.6. A Fourth-Order Scalar Equation [323] --
10.7. The First-Order Equation [328] --
10.8. Conjugate Point Problems [332] --
10.9. First-Order Equation with Parameter [336] --
11—From Differential to Integral Equations --
11.1. The Sturm-Liouville Case [339] --
11.2. Uniqueness and Existence of Solutions [341] --
11.3. Wronskian Identities [348] --
11.4. Variation of Parameters [350] --
11.5. Analytic Dependence on a Parameter [355] --
11.6. Eigenvalues and Orthogonality [356] --
11.7. Remarks on the Expansion Theorem [358] --
11.8. The Generalized First-Order Matrix Differential Equation [359] --
11.9. A Special Case [363] --
11.10. The Boundary Problem [364] --
12—Asymptotic Theory of Some Integral Equations --
12.1. Asymptotically Trigonometric Behavior [366] --
12.2. The -Function [371] --
12.3. A Non-Self-Adjoint Problem [375] --
12.4. The Sturm-Liouville Problem [381] --
12.5. Asymptotic Properties for the Generalization of y'' + [k2 + g(x)] y = 0 [384] --
12.6. Solutions of Integrable Square [391] --
12.7. Analytic Aspects of Asymptotic Theory [393] --
12.8. Approximations over a Finite Interval [398] --
12.9 Approximation to the Eigenfunctions [408] --
12.10. Completeness of the Eigenfunctions [411] --
Appendix I. Some Compactness Principles for Stieltjes Integrals --
1.1. Functions of Bounded Variation [416] --
1.2. The Riemann-Stieltjes Integral [418] --
1.3. A Convergence Theorem [423] --
1.4. The Helly-Bray Theorem [425] --
1.5. Infinite Interval and Bounded Integrand [426] --
1.6. Infinite Interval with Polynomial Integrand [428] --
1.7. A Periodic Case [430] --
1.8. The Matrix Extension [431] --
1.9. The Multi-Dimensional Case [434] --
Appendix II. Functions of Negative Imaginary Type --
II.1. Introduction [436] --
II.2. The Rational Case [437] --
II.3. Separation Property in the Meromorphic Case [439] --
Appendix III. Orthogonality of Vectors --
111.1. The Finite-Dimensional Case [441] --
111.2. The Infinite-Dimensional Case [442] --
Appendix IV. Some Stability Results for Linear Systems --
IV.1. A Discrete Case [447] --
IV.2. The Case of a Differential Equation [449] --
IV.3. A Second-Order Differential Equation [450] --
IV.4. The Mixed or Continuous-Discrete Case [452] --
IV.5, The Extended Gronwall Lemma [455] --
Appendix V. Eigenvalues of --
Varying Matrices --
V.l. Variational Expressions for Eigenvalues [457] --
V.2. Continuity and Monotonicity of Eigenvalues [459] --
V.3. A Further Monotonicity Criterion [461] --
V.4. Varying Unitary Matrices [464] --
V.5. Continuation of the Eigenvalues [465] --
V.6. Monotonicity of the Unit Circle [468] --
Appendix VI. Perturbation of Bases in Hilbert Space --
VI. 1. The Basic Result [471] --
VI.2. Continuous Variation of a Basis [473] --
VI. 3. Another Result [475] --
NOTATION AND TERMINOLOGY [476] --
LIST OF BOOKS AND MONOGRAPHS [478] --
MR, 31 #416
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