TY  - BOOK
AU  - Sakhnovich,Lev A.
ED  - SpringerLink (Online service)
TI  - Integral Equations with Difference Kernels on Finite Intervals: Second Edition, Revised and Extended 
T2  - Operator Theory: Advances and Applications,
SN  - 9783319164892
AV  - QA431 
U1  - 515.45 23
PY  - 2015///
CY  - Cham
PB  - Springer International Publishing, Imprint: Birkhäuser
KW  - mathematics
KW  - Integral equations
KW  - Operator theory
KW  - Probabilities
KW  - Mathematics
KW  - Integral Equations
KW  - Operator Theory
KW  - Probability Theory and Stochastic Processes
N1  - Preface to the second edition -- Introduction to the first edition -- 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.Levy Processes: Convolution-Type Form of the Infinitesimal Generator -- 8.On the Probability that the Levy Process (Class II) Remains within the Given Domain -- 9.Triangular Factorization and Cauchy Type Levy Processes -- 10.Levy Processes with Summable Levy Measures, Long Time Behavior -- 11.Open Problems -- Commentaries and Remarks -- Bibliography -- Glossary -- Index
N2  - 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
UR  - http://dx.doi.org/10.1007/978-3-319-16489-2
ER  -