Integral equations with difference kernels on finite intervals 2nd Edition by Lev Sakhnovich 3319164880 9781821871901

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Product details:

ISBN 10: 3319164880 

ISBN 13: 9781821871901

Author: Lev A. Sakhnovich

This book focuses on solving integral equations with difference kernels on finite intervals. The corresponding problem on the semiaxis was previously solved by N. Wiener–E. Hopf and by M.G. Krein. The problem on finite intervals, though significantly more difficult, may be solved using our method of operator identities. This method is also actively employed in inverse spectral problems, operator factorization and nonlinear integral equations. Applications of the obtained results to optimal synthesis, light scattering, diffraction, and hydrodynamics problems are discussed in this book, which also describes how the theory of operators with difference kernels is applied to stable processes and used to solve the famous M. Kac problems on stable processes. In this second edition these results are extensively generalized and include the case of all Levy processes. We present the convolution expression for the well-known Ito formula of the generator operator, a convolution expression that has proven to be fruitful. Furthermore we have added a new chapter on triangular representation, which is closely connected with previous results and includes a new important class of operators with non-trivial invariant subspaces. Numerous formulations and proofs have now been improved, and the bibliography has been updated to reflect more recent additions to the body of literature.

Table of contents:

1. Invertible Operator with a Difference Kernel

2. Equations of the First Kind with a Difference Kernel

3. Examples and Applications

4. Eigensubspaces and Fourier Transform

5. Integral Operators with-W Difference Kernels

6. Problems of Communication Theory

7. Lévy Processes: Convolution-type Form of the Infinitesimal Generator

8. On the Probability that the Lévy Process (Class II) Remains within the Given Domain

9. Triangular Factorization and Cauchy Type Lévy Processes

10. Lévy Processes with Summable Lévy Measures, Long Time Behavior

11. Open Problems

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