Operator Theory and Ill-Posed Problems

✍ Scribed by Mikhail M. (Jn.) Lavrent'ev; Lev Ja. Savel'ev

• Publisher: De Gruyter
• Year: 2011
• Tongue: English
• Leaves: 696
• Series: Inverse and Ill-Posed Problems Series; 50
• Category: Library

✦ Synopsis

This book consists of three major parts. The first two parts deal with general mathematical concepts and certain areas of operator theory. The third part is devoted to ill-posed problems. It can be read independently of the first two parts and presents a good example of applying the methods of calculus and functional analysis.

The first part "Basic Concepts" briefly introduces the language of set theory and concepts of abstract, linear and multilinear algebra. Also introduced are the language of topology and fundamental concepts of calculus: the limit, the differential, and the integral. A special section is devoted to analysis on manifolds. The second part "Operators" describes the most important function spaces and operator classes for both linear and nonlinear operators. Different kinds of generalized functions and their transformations are considered. Elements of the theory of linear operators are presented. Spectral theory is given a special focus. The third part "Ill-Posed Problems" is devoted to problems of mathematical physics, integral and operator equations, evolution equations and problems of integral geometry. It also deals with problems of analytic continuation.

Detailed coverage of the subjects and numerous examples and exercises make it possible to use the book as a textbook on some areas of calculus and functional analysis. It can also be used as a reference textbook because of the extensive scope and detailed references with comments.

✦ Table of Contents

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 Hubert 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 Monotonie 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. 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
9.6Volterra 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
12.1Statement 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