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Integral Equations with Difference Kernels on Finite Intervals de Lev A. Sakhnovich

Integral Equations with Difference Kernels on Finite Intervals: Operator Theory: Advances and Applications, cartea 84

Autor Lev A. Sakhnovich
en Limba Engleză Paperback – 26 sep 2011
Optimal synthesis, light scattering, and diffraction on a ribbon are just some of the applied problems for which integral equations with difference kernels are employed. The same equations are also met in important mathematical problems such as inverse spectral problems, nonlinear integral equations, and factorization of operators.
On the basis of the operator identity method, the theory of integral operators with difference kernels is developed here, and the connection with many applied and theoretical problems is studied. A number of important examples are analyzed.
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Specificații

ISBN-13: 9783034898560
ISBN-10: 3034898568
Pagini: 188
Ilustrații: 184 p.
Dimensiuni: 170 x 244 x 10 mm
Greutate: 0.31 kg
Ediția:Softcover reprint of the original 1st ed. 1996
Editura: Birkhäuser Basel
Colecția Birkhäuser
Seria Operator Theory: Advances and Applications

Locul publicării:Basel, Switzerland

Public țintă

Research

Cuprins

1. An Invertible Operator with a Difference Kernel.- §1. Constructing the inverse operator.- §2. Existence conditions and the structure of the inverse operator.- §3. Equations with a special right-hand side.- §4. Operators with a difference kernel in the space Lp(0,?).- §5. The use of the Fourier transform.- §6. Toeplitz matrices.- 2. Equations of the first kind with a Difference Kernel.- §1. Equations of the first kind with a special right-hand side.- §2. Solutions of equations of the first kind.- §3. Generalized solutions.- §4. On the behavior of solutions.- §5. On one class of integro-differential equations.- 3. Examples and Applications.- §1. Integral equations with kernels of power type.- §2. Integral equations with logarithmic type kernels.- §3. Regularization.- §4. Fractional integrals of purely imaginary order.- §5. On a class of integral equations which are solvable in exact form.- §6. On certain problems of hydrodynamics.- §7. Equations on the contact theory of elasticity.- §8. The equation of radiation transfer.- 4. Eigensubspaces and Fourier transform.- §1. Classification of eigensubspaces.- §2. On the distribution of the roots of Fourier images.- 5. Operator Bezoutiant and Roots of Entire Functions.- §1. Definition and properties of the operator B.- §2. Operator T corresponding to a pair of entire functions.- 6. Operator Identities and Systems of Equations with W-difference Kernel.- §1. The principal notions of S-knot theory.- §2. Systems with W -difference kernels.- §3. Prandtl equation.- 7. Integral Equations in the Theory of Stable Processes.- §1. The deduction of the integro-differential equations.- §2. Solution of the Kac problems.- §3. Two-sided estimation of the smallest eigenvalue of the operator.- 8. Problems ofCommunication Theory.- §1. Problem of optimal prediction.- §2. Problem of diffraction on a strip.- §3. Extremal problems in the theory of synthesis of antennae.- Commentaries and Remarks.- References.

Recenzii

“This monograph consists of 11 chapters, it is dedicated to the analysis of integral equations … . The monograph can be useful for researchers, undergraduate and graduate students in applied mathematics, whose research area is related to application of integral equations.” (Alexander N. Tynda, zbMATH 1334.45001, 2016)

Textul de pe ultima copertă

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 thathas 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.

Caracteristici

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 Includes supplementary material: sn.pub/extras