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.

Inhalt

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