Operator Theory and Ill-Posed Problems

Basic concepts
Chapter 1. Set theory
1.1. Sets
1.2. Correspondences
1.3. Relations
1.4. Induction
1.5. Natural numbers
Chapter 2. Algebra
2.1. Abstract algebra
2.2. Linear algebra
2.3. Multilinear algebra
Chapter 3. Calculus
3.1. Limit
3.2. Differential
3.3. Integral
3.4. Analysis on manifolds
Operators
Chapter 4. Linear operators
4.1. Hilbert spaces
4.2. Fourier series
4.3. Function spaces
4.4. Fourier transform
4.5. Bounded linear operators
4.6. Compact linear operators
4.7. Self-adjoint operators
4.8. Spectra of operators
4.9. Spectral theorem
4.10. Operator exponential
Chapter 5. Nonlinear operators
5.1. Fixed points
5.2. Saddle points
5.3. Monotonic operators
5.4. Nonlinear contractions
5.5. Degree theory
Ill-posed problems
Chapter 6. Classic problems
6.1. Mathematical description of the laws of physics
6.2. Equations of the first order
6.3. Classification of differential equations of the second order
6.4. Elliptic equations
6.5. Hyperbolic and parabolic equations
6.6. The notion of well-posedness
Chapter 7. Ill-posed problems
7.1. Ill-posed Cauchy problems
7.2. Analytic continuation and interior problems
7.3. Weakly and strongly ill-posed problems. Problems of differentiation
7.4. Reducing ill-posed problems to integral equations
Chapter 8. Physical problems leading to ill-posed problems
8.1. Interpretation of measurement data from physical devices
8.2. Interpretation of gravimetric data
8.3. Problems for the diffusion equation
8.4. Determining physical fields from the measurements data
8.5. Tomography
Chapter 9. Operator and integral equations
9.1. Definitions of well-posedness
9.2. Regularization
9.3. Linear operator equations
9.4. Integral equations with weak singularities
9.5. Scalar Volterra equations
9.6. Volterra operator equations
Chapter 10. Evolution equations
10.1. Cauchy problem and semigroups of operators
10.2. Equations in a Hilbert space
10.3. Equations with variable operator
10.4. Equations of the second order
10.5. Well-posed and ill-posed Cauchy problems
10.6. Equations with integro-differential operators
Chapter 11. Problems of integral geometry
11.1. Statement of problems of integral geometry
11.2. The Radon problem
11.3. Reconstructing a function from spherical means
11.4. Planar problem of the general form
11.5. Spatial problems of the general form
11.6. Problems of the Volterra type for manifolds invariant with
respect to the translation group
11.7. Planar problems of integral geometry with a perturbation
Chapter 12. Inverse problems
12.1. Statement of inverse problems
12.2. Inverse dynamic problem. A linearization method
12.3. A general method for studying inverse problems for hyperbolic
equations
12.4. The connection between inverse problems for hyperbolic,
elliptic, and parabolic equations
12.5. Problems of determining a Riemannian metric
Chapter 13. Several areas of the theory of ill-posed problems, inverse problems, and applications
Bibliography
Index