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Provides a new and effective method for solving integral equations with difference kernels
Uses the results obtained to investigate a number of theoretical and applied problems
Presents solutions to some well-known problems, in particular the M. Kac problems and a new form of the Levy-Ito equality
Studies a number of essential examples
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.
Related Subjects: Integral Equations, Operator Theory, Probability Theory and Stochastic Processes
Front Matter
Invertible Operator with a Difference Kernel
Equations of the First Kind with a Difference Kernel
Examples and Applications
Eigensubspaces and Fourier Transform
Integral Operators with W-Difference Kernels
Problems of Communication Theory
Lévy Processes: Convolution-type Form of the Infinitesimal Generator
On the Probability that the Lévy Process (Class II) Remains within the Given Domain
Triangular Factorization and Cauchy Type Lévy Processes
Lévy Processes with Summable Lévy Measures, Long Time Behavior
Open Problems
Back Matter