Integral geometry and convolution equations.

*(English)*Zbl 1043.53003
Dordrecht: Kluwer Academic Publishers (ISBN 1-4020-1628-X/hbk). xii, 454 p. (2003).

The book under review reflects the modern state of the results and is mainly based on the results of the author. The book consists of five parts. Part 1 (chapters 1–8) contains all necessary preliminaries (distributions, classes of functions, special functions and spherical harmonics, the Fourier transform etc). Part 2 (chapters 1–3) is connected with the description of some classes of functions with zero integrals over balls of a fixed radius. Hyperbolic and spherical analogues of such classes are also considered and their properties are studied. In addition, the local version of the two radii theorem is given. It is connected with the following problem: For what sets \(E\) of positive numbers and \(f(x)\in L_{loc} (\mathbb R^n)\) the assumption that for all \(r\in E\) and \(x\in \mathbb R^n\) the equality
\[
\int\limits_{| u| \leq r} f(x+u)du=0
\]
implies that \(f(x)\equiv 0\).

Part 3 (chapters 1–4) is devoted to solvability and uniqueness problems of convolution equations \((f\ast\varphi)(x)=0\) on domains in \(\mathbb R^1\) and \(\mathbb R^n\). In the last case the equations (and the system of such equations) are considered on the domains with spherical symmetry. Except this the author studies the behavior of solutions of convolution equations at infinity and describes solutions with growth restriction at infinity.

In Part 4 (chapters 1–7) the Pompeiu problem is studied in explicit form and the sets with Pompeiu property are described. Let \(M(n)\) be the group of Euclidean motions in \(\mathbb R^n\). We recall that a compact set \(A\in \mathbb R^n, n\geq 2\) has the Pompeiu property if for \(f(x)\in L_{loc} (\mathbb R^n)\) the assumption that the condition \[ \int\limits_{\lambda A} f(x)\,dx=0 \eqno(1) \] for all \(\lambda\in M(n)\) implies that \(f(x)\equiv 0.\) Such set \(A\) is a Pompeiu set. It is known that if \(A\) is a ball in \(\mathbb R^n\) then \(A\) is not a Pompeiu set, while if \(A=[0,1]^n\) is a cube (or ellipsoid, which is not a ball) in \(\mathbb R^n\) then \(A\) have the Pompeiu property. The criterium for a set \(A\in \mathbb R^n\) to have the Pompeiu property is given in terms of the Fourier transform of the characteristic function of the set \(A\). The local Pompeiu property, corresponding to the case when the function \(f\) is defined on a bounded domain in \(\mathbb R^n\) is also considered. The problem of minimal radius of a ball on which \(A\) is a Pompeiu set is studied. In particular, such problem is considered for the cases when \(A\) is a parallelepiped, polyhedron or ellipsoid.

Part 5 (chapters 1–8) is devoted to various interesting applications. All parts contain interesting examples and a special chapter with open and unsolved problems.

The book is written in a very clear manner and should be useful both for experts in the field and for postgraduate students. The wide list of citations includes more than 250 items.

Contents. Part 1. Preliminaries. Chapter 1. Set and mappings. Chapter 2. Some classes of functions. Chapter 3. Distributions. Chapter 4. Some special functions. Chapter 5. Some results related to spherical harmonics. Chapter 6. Fourier transform and related questions. Chapter 7. Partial differential equations. Chapter 8. Radon transform over hyperplanes. Chapter 9. Comments and open problems.

Part 2. Functions with zero integrals over balls of a fixed radius. Chapter 1. Functions with zero averages over balls on subsets of the space \(\mathbb R^n\). Chapter 2. Averages over balls on hyperbolic spaces. Chapter 3. Functions with zero integrals over spherical caps. Chapter 4. Comments and open problems.

Part 3. Convolution equation on domains in \(\mathbb R^n\). Chapter 1. One-dimensional case. Chapter 2. General solution of convolution equation in domain with spherical symmetry. Chapter 3. Behavior of solutions of convolution equation at infinity. Chapter 4. Systems of convolution equations. Chapter 5. Comments and open problems.

Part 4. Extremal versions of the Pompeiu problem. Chapter 1. Sets with the Pompeiu property. Chapter 2. Functions with vanishing integrals over parallelepipeds. Chapter 3. Polyhedra with local Pompeiu property. Chapter 4. Functions with vanishing integrals over ellipsoids. Chapter 5. Other sets with local Pompeiu property. Chapter 6. The ’three squares’ problem and related questions. Chapter 7. Injectivity sets of the Pompeiu transform. Chapter 8. Comments and open problems.

Part 5. First applications and related questions. Chapter 1. Injectivity sets for spherical Radon transform. Chapter 2. Some questions of approximation theory. Chapter 3. Cap theorems. Chapter 4. Morera type theorems. Chapter 5. Mean value characterization of various classes of functions. Chapter 6. Applications to partial differential equations. Chapter 7. Some questions of measure theory. Chapter 8. Functions with zero integrals in problems of the discrete geometry. Chapter 9. Comments and open problems.

Bibliography. Author index. Subject index. Basic notations.

Part 3 (chapters 1–4) is devoted to solvability and uniqueness problems of convolution equations \((f\ast\varphi)(x)=0\) on domains in \(\mathbb R^1\) and \(\mathbb R^n\). In the last case the equations (and the system of such equations) are considered on the domains with spherical symmetry. Except this the author studies the behavior of solutions of convolution equations at infinity and describes solutions with growth restriction at infinity.

In Part 4 (chapters 1–7) the Pompeiu problem is studied in explicit form and the sets with Pompeiu property are described. Let \(M(n)\) be the group of Euclidean motions in \(\mathbb R^n\). We recall that a compact set \(A\in \mathbb R^n, n\geq 2\) has the Pompeiu property if for \(f(x)\in L_{loc} (\mathbb R^n)\) the assumption that the condition \[ \int\limits_{\lambda A} f(x)\,dx=0 \eqno(1) \] for all \(\lambda\in M(n)\) implies that \(f(x)\equiv 0.\) Such set \(A\) is a Pompeiu set. It is known that if \(A\) is a ball in \(\mathbb R^n\) then \(A\) is not a Pompeiu set, while if \(A=[0,1]^n\) is a cube (or ellipsoid, which is not a ball) in \(\mathbb R^n\) then \(A\) have the Pompeiu property. The criterium for a set \(A\in \mathbb R^n\) to have the Pompeiu property is given in terms of the Fourier transform of the characteristic function of the set \(A\). The local Pompeiu property, corresponding to the case when the function \(f\) is defined on a bounded domain in \(\mathbb R^n\) is also considered. The problem of minimal radius of a ball on which \(A\) is a Pompeiu set is studied. In particular, such problem is considered for the cases when \(A\) is a parallelepiped, polyhedron or ellipsoid.

Part 5 (chapters 1–8) is devoted to various interesting applications. All parts contain interesting examples and a special chapter with open and unsolved problems.

The book is written in a very clear manner and should be useful both for experts in the field and for postgraduate students. The wide list of citations includes more than 250 items.

Contents. Part 1. Preliminaries. Chapter 1. Set and mappings. Chapter 2. Some classes of functions. Chapter 3. Distributions. Chapter 4. Some special functions. Chapter 5. Some results related to spherical harmonics. Chapter 6. Fourier transform and related questions. Chapter 7. Partial differential equations. Chapter 8. Radon transform over hyperplanes. Chapter 9. Comments and open problems.

Part 2. Functions with zero integrals over balls of a fixed radius. Chapter 1. Functions with zero averages over balls on subsets of the space \(\mathbb R^n\). Chapter 2. Averages over balls on hyperbolic spaces. Chapter 3. Functions with zero integrals over spherical caps. Chapter 4. Comments and open problems.

Part 3. Convolution equation on domains in \(\mathbb R^n\). Chapter 1. One-dimensional case. Chapter 2. General solution of convolution equation in domain with spherical symmetry. Chapter 3. Behavior of solutions of convolution equation at infinity. Chapter 4. Systems of convolution equations. Chapter 5. Comments and open problems.

Part 4. Extremal versions of the Pompeiu problem. Chapter 1. Sets with the Pompeiu property. Chapter 2. Functions with vanishing integrals over parallelepipeds. Chapter 3. Polyhedra with local Pompeiu property. Chapter 4. Functions with vanishing integrals over ellipsoids. Chapter 5. Other sets with local Pompeiu property. Chapter 6. The ’three squares’ problem and related questions. Chapter 7. Injectivity sets of the Pompeiu transform. Chapter 8. Comments and open problems.

Part 5. First applications and related questions. Chapter 1. Injectivity sets for spherical Radon transform. Chapter 2. Some questions of approximation theory. Chapter 3. Cap theorems. Chapter 4. Morera type theorems. Chapter 5. Mean value characterization of various classes of functions. Chapter 6. Applications to partial differential equations. Chapter 7. Some questions of measure theory. Chapter 8. Functions with zero integrals in problems of the discrete geometry. Chapter 9. Comments and open problems.

Bibliography. Author index. Subject index. Basic notations.

Reviewer: Nikolai K. Karapetyants (Rostov-na-Donu)

##### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C65 | Integral geometry |

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

44A12 | Radon transform |