Linear Integral Equations

1. Normed Spaces.- 1.1 Convergence and Continuity.- 1.2 Open and Closed Sets.- 1.3 Completeness.- 1.4 Compactness.- 1.5 Scalar Products.- 1.6 Best Approximation.- 2. Bounded and Compact Operators.- 2.1 Bounded Operators.- 2.2 Integral Operators.- 2.3 Neumann Series.- 2.4 Compact Operators.- 3. The Riesz Theory.- 3.1 Riesz Theory for Compact Operators.- 3.2 Spectral Theory for Compact Operators.- 3.3 Volterra Integral Equations.- 4. Dual Systems and Fredholm Theory.- 4.1 Dual Systems Via Bilinear Forms.- 4.2 Dual Systems Via Sesquilinear Forms.- 4.3 Positive Dual Systems.- 4.4 The Fredholm Alternative.- 4.5 Boundary Value Problems.- 5. Regularization in Dual Systems.- 5.1 Regularizers.- 5.2 Normal Solvability.- 5.3 Index.- 6. Potential Theory.- 6.1 Harmonic Functions.- 6.2 Boundary Value Problems: Uniqueness.- 6.3 Surface Potentials.- 6.4 Boundary Value Problems: Existence.- 6.5 Supplements.- 7. Singular Integral Equations.- 7.1 Holder Continuity.- 7.2 The Cauchy Integral Operator.- 7.3 The Riemann Problem.- 7.4 Singular Integral Equations with Cauchy Kernel.- 7.5 Cauchy Integral and Logarithmic Potential.- 7.6 Supplements.- 8. Sobolev Spaces.- 8.1 Fourier Expansion.- 8.2 The Sobolev Space Hp[0, 2?].- 8.3 The Sobolev Space Hp[?].- 8.4 Weak Solutions to Boundary Value Problems.- 9. The Heat Equation.- 9.1 Initial Boundary Value Problem: Uniqueness.- 9.2 Heat Potentials.- 9.3 Initial Boundary Value Problem: Existence.- 10. Operator Approximations.- 10.1 Approximations Based on Norm Convergence.- 10.2 Uniform Boundedness Principle.- 10.3 Collectively Compact Operators.- 10.4 Approximations Based on Pointwise Convergence.- 10.5 Successive Approximations.- 11. Degenerate Kernel Approximation.- 11.1 Finite Dimensional Operators.- 11.2 Degenerate Kernels Via Interpolation.- 11.3 Degenerate Kernels Via Expansions.- 12. Quadrature Methods.- 12.1 Numerical Integration.- 12.2 Nyström’s Method.- 12.3 Nyström’s Method for Weakly Singular Kernels.- 13. Projection Methods.- 13.1 The Projection Method.- 13.2 The Collocation Method.- 13.3 The Galerkin Method.- 14. Iterative Solution and Stability.- 14.1 The Method of Residual Correction.- 14.2 Multi-Grid Methods.- 14.3 Stability of Linear Systems.- 15. Equations of the First Kind.- 15.1 Ill-Posed Problems.- 15.2 Regularization of Ill-Posed Problems.- 15.3 Compact Self Adjoint Operators.- 15.4 Singular Value Decomposition.- 15.5 Regularization Schemes.- 16. Tikhonov Regularization.- 16.1 The Tikhonov Functional.- 16.2 Weak Convergence.- 16.3 Quasi-Solutions.- 16.4 Minimum Norm Solutions.- 16.5 Classical Tikhonov Regularization.- 17. Regularization by Discretization.- 17.1 Projection Methods for Ill-Posed Equations.- 17.2 The Moment Method.- 17.3 Hilbert Spaces with Reproducing Kernel.- 17.4 Moment Collocation.- 18. Inverse Scattering Theory.- 18.1 Ill-Posed Integral Equations in Potential Theory.- 18.2 An Inverse Acoustic Scattering Problem.- 18.3 Numerical Methods in Inverse Scattering.