Linear Integral Equations

1 Normed Spaces.- 1.1 Convergence and Continuity.- 1.2 Completeness.- 1.3 Compactness.- 1.4 Scalar Products.- 1.5 Best Approximation.- Problems.- 2 Bounded and Compact Operators.- 2.1 Bounded Operators.- 2.2 Integral Operators.- 2.3 Neumann Series.- 2.4 Compact Operators.- Problems.- 3 Riesz Theory.- 3.1 Riesz Theory for Compact Operators.- 3.2 Spectral Theory for Compact Operators.- 3.3 Volterra Integral Equations.- Problems.- 4 Dual Systems and Fredholm Alternative.- 4.1 Dual Systems via Bilinear Forms.- 4.2 Dual Systems via Sesquilinear Forms.- 4.3 The Fredholm Alternative.- 4.4 Boundary Value Problems.- Problems.- 5 Regularization in Dual Systems.- 5.1 Regularizers.- 5.2 Normal Solvability.- 5.3 Index.- Problems.- 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 Nonsmooth Boundaries.- Problems.- 7 Singular Integral Equations.- 7.1 Hölder Continuity.- 7.2 The Cauchy Integral Operator.- 7.3 The Riemann Problem.- 7.4 Integral Equations with Cauchy Kernel.- 7.5 Cauchy Integral and Logarithmic Potential.- 7.6 Logarithmic Single-Layer Potential on an Arc.- Problems.- 8 Sobolev Spaces.- 8.1 The Sobolev Space Hp[0, 2?].- 8.2 The Sobolev Space Hp(?).- 8.3 Weak Solutions to Boundary Value Problems.- Problems.- 9 The Heat Equation.- 9.1 Initial Boundary Value Problem: Uniqueness.- 9.2 Heat Potentials.- 9.3 Initial Boundary Value Problem: Existence.- Problems.- 10 Operator Approximations.- 10.1 Approximations via Norm Convergence.- 10.2 Uniform Boundedness Principle.- 10.3 Collectively Compact Operators.- 10.4 Approximations via Pointwise Convergence.- 10.5 Successive Approximations.- Problems.- 11 Degenerate Kernel Approximation.- 11.1 Degenerate Operators and Kernels.- 11.2 Interpolation.- 11.3 Trigonometric Interpolation.- 11.4 Degenerate Kernels via Interpolation.- 11.5 Degenerate Kernels via Expansions.- Problems.- 12 Quadrature Methods.- 12.1 Numerical Integration.- 12.2 Nyström's Method.- 12.3 Weakly Singular Kernels.- 12.4 Nyström's Method in Sobolev Spaces.- Problems.- 13 Projection Methods.- 13.1 The Projection Method.- 13.2 Projection Methods for Equations of the Second Kind.- 13.3 The Collocation Method.- 13.4 Collocation Methods for Equations of the First Kind.- 13.5 The Galerkin Method.- Problems.- 14 Iterative Solution and Stability.- 14.1 Stability of Linear Systems.- 14.2 Two-Grid Methods.- 14.3 Multigrid Methods.- 14.4 Fast Matrix-Vector Multiplication.- Problems.- 15 Equations of the First Kind.- 15.1 Ill-Posed Problems.- 15.2 Regularization of 1ll-Posed Problems.- 15.3 Compact Self-Adjoint Operators.- 15.4 Singular Value Decomposition.- 15.5 Regularization Schemes.- Problems.- 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.- Problems.- 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.- Problems.- 18 Inverse Boundary Value Problems.- 18.1 Ill-Posed Equations in Potential Theory.- 18.2 An Inverse Problem in Potential Theory.- 18.3 Approximate Solution via Potentials.- 18.4 Differentiability with Respect to the Boundary.- Problems.- References.