# Integral Equations with Difference Kernels on Finite Intervals (2nd ed.)

This is a revised and extended revision of the previous edition from 1996 that appeared as volume 84 of the series *Operator Theory Advances and Applications* (OTAA). The chapter on operator Bezoutiants and roots of entire functions has been removed since it was treated in another of Sakhnoviches books: *Lévy Processes, Integral Equations, Statistical Physics: Connections and Interactions* (2012, OTAA vol. 225). On the other hand, the part on Lévy processes is largely extended in the form of new chapters. Also a number of open problems has been added.

As the title says, the book considers the solution of the equation $Sf=\varphi$ with $Sf=\sum_{j=M}^N \mu_j f(x−x_j)+\int_0^\omega k(x−t)f(t)dt$ with as a special case $\mu_j=0$ for $j\ne0$ and $x_0=0$. By modifying the kernel, this can be transformed into

\[ Sf=\frac{d}{dx}\int_0^\omega s(x−t)f(t)dt \quad\mbox{where}\quad \left\{\begin{array}{ll}s(x)=\int_0^x k(u)du+\mu_+,& x>0\\ s(x)=\int_0^x k(u)du+\mu_−,& x<0\end{array}\right. \quad\mbox{and}\quad \mu_++\mu_−=\mu. \]

This relies on a relation like $S\frac{d}{dx}=\frac{d}{dx}S$, which is a special case of more general operator identities of the form $AS−SB=Q$. It are the identities of this kind that are used in this book to solve the equation, i.e., to construct an inverse $T=S^{-1}$ which obviously has to satisfy $TA-BT=TQT$. For example when $Af=i\int_0^x f(t)dt$ and $A^∗f=−i\int_x^\omega f(t)dt$ then $(AS-SA^∗)f=i\int_0^\omega(M(t)+N(t))f(t)dt$ with $M(x)=s(x)$ and $N(x)=-s(-x)$. It then follows that the knowledge of $N_1$ and $N_2$ satisfying $SN_1=M$ and $SN_2=\mathbb{1}$ allows to find the structure of $T$ or even construct $T$ explicitly in some cases.

A special role is also played by a right-hand side equal to $e^{i\lambda x}$ with solution $B(x,\lambda)$. This allows to make a link with scattering theory since the function $\rho(\lambda,\mu)=\int_0^\omega B(x,\lambda)e^{i\mu x}dx$ has the interpretation of a reflection coefficient. The link with roots of entire functions can be made because the operator $T$ is the analogue of a Bezoutiant, which, in the classical case is used to find common zeros of two polynomials.

Several applications sometimes come to the forefront like hydrodynamics, radiation, communication and antennas. Other applications like elasticity and diffraction lead to even more general situations where the one interval $[0,\omega]$ is replaced by a system of intervals. The physical interpretation of the results is sometimes given but not always.

Different cases are discussed in successive chapters like difference kernels belonging to different $L^p$-spaces or with a power of logarithmic behavior with and without particular right-hand sides. As we mentioned already, Lévy processes take a particular place in this second edition (chapters 7-10). This is basically a stochastic process with zero initial condition and with independent and stationary increments. The transition operators form a continuous semigroup and it is shown that its infinitesimal generator is of convolution type. By introducing a quasi potential an analysis is made of the process staying within a certain domain when the measure is not integrable. Several examples are given for triangular factorization when $S$ is a positive definite operator. Finally, if the measure of the process is integrable, the quasi potential takes a particular form and again it is analyzed when the process will stay within a domain either after a finite number of steps or asymptotically.

The coherence of the different chapters is not very strong. They give results for variants and particular cases of the equations that all fall under the title of the book, but that are somewhat independent from each other. Notwithstanding the practical applicability and the importance of these convolution type equations for engineers and applied sciences, the applications are not the prime interest of this book. Numerical and computational aspects are not considered at all. This is a theoretical study of these equations anchored on a strong Russian tradition referring to problems investigated by Krein, Kac and others. Also the list of references consists mainly of items from the Russian literature. For the mathematician, the open problems at the end of the book may be some good challenges in operator theory or stochastic processes to consider.

Let me mention that at about the same time of this second edition, another volume was published in the OTAA series as volume 244 *Recent Advances in Inverse Scattering, Schur Analysis and Stochastic Processes* with 10 research papers dedicated to Lev Sakhnovich on the occasion of his 80th birthday. It also contains some biographical notes and a list of his publications.

**Submitted by Adhemar Bultheel |

**20 / Jul / 2015