##
**Linear integral equations.**
*(English)*
Zbl 0671.45001

Applied Mathematical Sciences, 82. Berlin etc.: Springer-Verlag. xi, 299 p. (1989).

This book is intended to serve as an introductory text for students and engineers interested in the theory, applications and numerical methods for solving integral equations. It consists of 18 chapters, a bibliography and an index. At the end of each chapter there are some exercises.

The theory of integral equations in this text includes the Fredholm-Riesz theory, a result on Volterra equations with continuous kernels on compact intervals, and Noether’s theorems for one-dimensional singular integral equations. This material was already available in the textbook literature. Applications of integral equations are represented by a sketch of classical potential theory for Laplace’s and heat equations. Properties of potentials of singular and double layers are stated without proofs.

Numerical methods are represented by quadrature, projection and iterative methods, a discussion of ill-posed problems and methods for their regularization. An application to inverse scattering problems is considered. The problem consists of finding the boundary of a reflecting obstacle from the scattering amplitude.

In an introductory text the reader could expect to find: 1) more examples of applications (estimation theory, formation of the image by optical instruments, laser theory, antenna synthesis, to mention just a few examples); 2) explicitly solvable classes of integral equations (convolution kernels, Wiener-Hopf equations, etc.); 3) bibliographical remarks and a list of basic texts on the subject published earlier.

These expectations are not met: 1) explicitly solvable classes of integral equations are not even mentioned; 2) most of the texts on integral equations are not mentioned (e.g., Goursat, Tricomi, Petrovski, Courant-Hilbert, Hochstadt, Mikhlin, etc.), an exception being Pogorzelski’s book; monographs by Muskhelishvili, Prössdorf, Mikhlin and Prössdorf, on singular integral equations are mentioned; no results on the distributions of eigenvalues of integral equations are mentioned; the classical Fredholm results on the representation of the resolvent of integral operators are not mentioned; integral equations with non- negative kernels are not discussed, etc. A well known reference book on integral equations [P. Zabreiko et al., Integral equations, Nauka, Moscow (1968)] as well as many monograhs on the topics discussed in the book (projection methods, potential theory, iterative methods, ill-posed problems etc., are not mentioned (Gohberg-Feldman; G. Vainikko; V. V. Ivanov; S. G. Krein; Clancey-Gohberg; C. Nedelec; Cherski-Gahov; V. K. Ivanov, V. Tanana, V. Vasin; Kantorovich-Akilov; these are a few examples of the authors of monographs on the topics discussed in the book under review whose books are not mentioned in the bibliography); as a result, the reader will have some difficulties in getting a proper perspective on the subject.

On p. 33 a bounded kernel is called weakly singular. This is unusual terminology which is not in agreement with the standard terminology on p. 175. In the last formula of section 8.1, p. 109, the coefficient \(\pi\) /12 should stand in place of \(\pi\) /6.

On the positive side, the book contains elements of much of the material which is discussed at depth in a more advanced literature. This may be useful for students and engineers who want to use integral equations. The presentation is clear. The book can be useful for many readers.

The theory of integral equations in this text includes the Fredholm-Riesz theory, a result on Volterra equations with continuous kernels on compact intervals, and Noether’s theorems for one-dimensional singular integral equations. This material was already available in the textbook literature. Applications of integral equations are represented by a sketch of classical potential theory for Laplace’s and heat equations. Properties of potentials of singular and double layers are stated without proofs.

Numerical methods are represented by quadrature, projection and iterative methods, a discussion of ill-posed problems and methods for their regularization. An application to inverse scattering problems is considered. The problem consists of finding the boundary of a reflecting obstacle from the scattering amplitude.

In an introductory text the reader could expect to find: 1) more examples of applications (estimation theory, formation of the image by optical instruments, laser theory, antenna synthesis, to mention just a few examples); 2) explicitly solvable classes of integral equations (convolution kernels, Wiener-Hopf equations, etc.); 3) bibliographical remarks and a list of basic texts on the subject published earlier.

These expectations are not met: 1) explicitly solvable classes of integral equations are not even mentioned; 2) most of the texts on integral equations are not mentioned (e.g., Goursat, Tricomi, Petrovski, Courant-Hilbert, Hochstadt, Mikhlin, etc.), an exception being Pogorzelski’s book; monographs by Muskhelishvili, Prössdorf, Mikhlin and Prössdorf, on singular integral equations are mentioned; no results on the distributions of eigenvalues of integral equations are mentioned; the classical Fredholm results on the representation of the resolvent of integral operators are not mentioned; integral equations with non- negative kernels are not discussed, etc. A well known reference book on integral equations [P. Zabreiko et al., Integral equations, Nauka, Moscow (1968)] as well as many monograhs on the topics discussed in the book (projection methods, potential theory, iterative methods, ill-posed problems etc., are not mentioned (Gohberg-Feldman; G. Vainikko; V. V. Ivanov; S. G. Krein; Clancey-Gohberg; C. Nedelec; Cherski-Gahov; V. K. Ivanov, V. Tanana, V. Vasin; Kantorovich-Akilov; these are a few examples of the authors of monographs on the topics discussed in the book under review whose books are not mentioned in the bibliography); as a result, the reader will have some difficulties in getting a proper perspective on the subject.

On p. 33 a bounded kernel is called weakly singular. This is unusual terminology which is not in agreement with the standard terminology on p. 175. In the last formula of section 8.1, p. 109, the coefficient \(\pi\) /12 should stand in place of \(\pi\) /6.

On the positive side, the book contains elements of much of the material which is discussed at depth in a more advanced literature. This may be useful for students and engineers who want to use integral equations. The presentation is clear. The book can be useful for many readers.

Reviewer: A.G.Ramm

### MSC:

45-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to integral equations |

65R20 | Numerical methods for integral equations |

45A05 | Linear integral equations |

45B05 | Fredholm integral equations |

45E05 | Integral equations with kernels of Cauchy type |

47B15 | Hermitian and normal operators (spectral measures, functional calculus, etc.) |

45L05 | Theoretical approximation of solutions to integral equations |

65J10 | Numerical solutions to equations with linear operators |

31B20 | Boundary value and inverse problems for harmonic functions in higher dimensions |

74J25 | Inverse problems for waves in solid mechanics |