Integral Geometry and Convolution Equations

1. Preliminaries.- 1 Sets and Mappings.- 2 Some Classes of Functions.- 3 Distributions.- 4 Some Special Functions.- 5 Some Results Related to Spherical Harmonics.- 6 Fourier Transform and Related Questions.- 7 Partial Differential Equations.- 8 Radon Transform Over Hyperplanes.- 9 Comments And Open Problems.- 2. Functions with zero integrals over balls of a fixed radius.- 1 Functions With Zero Averages Over Balls on Subsets of The Space ?n.- 2 Averages Over Balls on Hyperbolic Spaces.- 3 Functions With Zero Integrals Over Spherical Caps.- 4 Comments And Open Problems.- 3. Convolution equation on domains in ?n.- 1 One-Dimensional Case.- 2 General Solution Of Convolution Equation In Domains With Spherical Symmetry.- 3 Behavior of Solutions of Convolution Equation at Infinity.- 4 Systems Of Convolution Equations.- 5 Comments And Open Problems.- 4. Extremal versions of the Pompeiu problem.- 1 Sets With The Pompeiu Property.- 2 Functions With Vanishing Integrals Over Parallelepipeds.- 3 Polyhedra With Local Pompeiu Property.- 4 Functions with Vanishing Integrals over Ellipsoids.- 5 Other Sets With Local Pompeiu Property.- 6 The 'Three Squares' Problem And Related Questions.- 7 Injectivity Sets of the Pompeiu Transform.- 8 Comments and Open Problems.- 5. First applications and related questions.- 1 Injectivity Sets For Spherical Radon Transform.- 2 Some Questions Of Approximation Theory.- 3 Gap Theorems.- 4 Morera Type Theorems.- 5 Mean Value Characterization of Various Classes of Functions.- 6 Applications To Partial Differential Equations.- 7 Some Questions Of Measure Theory.- 8 Functions With Zero Integrals In Problems Of The Discrete Geometry.- 9 Comments And Open Problems.- Author Index.- Basic Notations.

From the reviews:



"The book under review reflects the modern state of the results and is mainly based on the results of the author. ... 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." (Nikolai K. Karapetyants, Zentralblatt MATH, Vol. 1043 (18), 2004)

"The book is devoted to the problem of injectivity for convolution operators of geometric nature. ... A survey of works in the area by other authors is presented as well. The monograph contains a collection of interesting and original results ... . The book will be of interest for specialists in analysis, in particular, in harmonic analysis, spectral theory, invariant function spaces and integral equations. It may serve as a source for further research in the area." (Mark Agranovsky, Mathematical Reviews, Issue 2005 e)