| Table of Contents | p. v |
| Preface | p. xi |
| Preliminaries | p. 1 |
| Sets and Mappings | p. 1 |
| Some concepts related to sets | p. 1 |
| Mappings | p. 3 |
| Some Classes of Functions | p. 5 |
| Integrable functions | p. 5 |
| Spaces of smooth functions | p. 6 |
| Quasi-analytic classes of functions | p. 8 |
| Entire functions | p. 9 |
| Distributions | p. 12 |
| Simplest properties of distributions | p. 12 |
| Tensor product and convolution | p. 13 |
| Regularization | p. 15 |
| Some Special Functions | p. 16 |
| Gamma function | p. 16 |
| Hypergeometric function | p. 16 |
| The Legendre functions | p. 17 |
| The Gegenbauer polynomials | p. 19 |
| The cylindrical functions | p. 21 |
| Some Results Related to Spherical Harmonics | p. 26 |
| Simplest properties of spherical harmonics | p. 26 |
| Expansions over spherical harmonics | p. 28 |
| The functions [Phi superscript k,l subscript z,eta], [Psi superscript k,l subscript z,eta] | p. 32 |
| Some properties of solutions of convolution equations | p. 34 |
| Fourier Transform and Related Questions | p. 37 |
| The Fourier transform of integrable functions | p. 37 |
| The Fourier-Laplace transform | p. 39 |
| Analytic wave front set | p. 43 |
| Partial Differential Equations | p. 46 |
| Some general results | p. 46 |
| The equation ([Delta] + [lambda superscript 2])[superscript m]f = 0 | p. 47 |
| Radon Transform Over Hyperplantes | p. 49 |
| Simplest properties of the Radon transform over hyperplanes | p. 49 |
| Functions with vanishing integrals over hyperplanes | p. 50 |
| Comments and Open Problems | p. 55 |
| Functions with zero integrals over balls of a fixed radius | p. 57 |
| Functions with Zero Averages Over Balls on Subsets of the Space R[superscript n] | p. 57 |
| Examples of functions with zero averages over balls and their simplest properties | p. 57 |
| Uniqueness theorems | p. 61 |
| Description of functions in the class V[subscript r](B[subscript R]) | p. 65 |
| A local two-radii theorem | p. 69 |
| A local one-radius theorem | p. 81 |
| Description of functions in the class V[subscript r](B[subscript a,b]) | p. 86 |
| [infinity]-radii theorem | p. 89 |
| The support problem for averages over balls | p. 91 |
| Functions with zero integrals over balls on half-space | p. 95 |
| Averages Over Balls on Hyperbolic Spaces | p. 100 |
| Auxiliary constructions related to hyperbolic space | p. 100 |
| Hyperbolic analogues of the classes V[subscript r](B[subscript R]) | p. 108 |
| Uniqueness theorem and description of the class V[subscript r](B[subscript R]) in the hyperbolic case | p. 111 |
| A local two-radii theorem on hyperbolic space | p. 115 |
| Functions with Zero Integrals Over Spherical Caps | p. 122 |
| Some auxiliary assertions | p. 122 |
| Spherical analogues of the classes V[subscript r](B[subscript R]) | p. 127 |
| Theorems on averages over spherical caps | p. 131 |
| Comments and Open Problems | p. 137 |
| Convolution equation on domains in R[superscript n] | p. 143 |
| One-Dimensional Case | p. 143 |
| Auxiliary constructions | p. 143 |
| Partial solutions of convolution equation | p. 148 |
| Uniqueness theorem | p. 150 |
| General solution of convolution equation | p. 152 |
| Solutions of convolution equations with growth restrictions at infinity | p. 157 |
| Systems of convolution equations | p. 159 |
| Generalizations of local two-radii theorem in one-dimensional case | p. 162 |
| General Solution of Convolution Equation in Domains with Spherical Symmetry | p. 169 |
| Auxiliary constructions | p. 169 |
| Partial solutions of convolution equation | p. 172 |
| Uniqueness theorems | p. 176 |
| General solution of convolution equation in a ball | p. 180 |
| General solution of convolution equation in a spherical annulus | p. 185 |
| Theorem on a removable singularity for solutions of convolution equation | p. 189 |
| Behavior of Solutions of Convolution Equation at Infinity | p. 191 |
| Convolution equation on R[superscript n], n [greater than or equal] 2 | p. 191 |
| Convolution equation on exterior of compact set | p. 196 |
| Convolution equation on half-space | p. 199 |
| Systems of Convolution Equations | p. 201 |
| Systems of convolution equations on domains with spherical symmetry | p. 201 |
| Generalizations of local two-radii theorem | p. 204 |
| Comments and Open Problems | p. 212 |
| Extremal versions of the Pompeiu problem | p. 214 |
| Sets with the Pompeiu Property | p. 214 |
| Pompeiu problem | p. 214 |
| The local Pompeiu property | p. 219 |
| Upper and lower estimates for R(A) | p. 223 |
| Functions with Vanishing Integrals Over Parallelepipeds | p. 226 |
| Statement of central result | p. 226 |
| A functional equation | p. 227 |
| Estimates of some quantities related to [zeta] | p. 230 |
| Examples of functions in the class B(A, B[subscript r]) | p. 233 |
| Properties of functions in the class B(A, B[subscript r]) | p. 237 |
| Proof of Theorem 2.1 | p. 246 |
| Polyhedra with Local Pompeiu Property | p. 250 |
| A functional equation | p. 250 |
| Upper estimate R(A) for polyhedra | p. 255 |
| Value R(A) for polygons | p. 259 |
| Value R(A) for convex polyhedra | p. 264 |
| The regular polyhedra | p. 269 |
| Functions with Vanishing Integrals Over Ellipsoids | p. 271 |
| Integration over ellipsoids with a common center | p. 271 |
| Some auxiliary results on dispositions of ellipsoids included to the ball | p. 278 |
| Existence result for solutions of auxiliary integral equation | p. 282 |
| Examples of functions with zero integrals over ellipsoids | p. 287 |
| Functions with zero integrals over congruent ellipsoids | p. 292 |
| Value R(E[subscript a]) for ellipsoid which is not a ball | p. 298 |
| Other Sets with Local Pompeiu Property | p. 303 |
| Value R(A) for a half-ball | p. 303 |
| Value R(A) for circular sector and Reuleaux triangle | p. 306 |
| The 'Three Squares' Problem and Related Questions | p. 311 |
| Statement of the problem and some generalizations | p. 311 |
| Local versions of the 'three squares' problem | p. 313 |
| Some hyperbolic analogues | p. 316 |
| Injectivity Sets of the Pompeiu Transform | p. 320 |
| Main problems | p. 320 |
| Sufficient conditions for injectivity P[subscript F] | p. 321 |
| Characterization of injectivity sets of P[subscript F] for some F | p. 323 |
| Comments and Open Problems | p. 334 |
| First applications and related questions | p. 339 |
| Injectivity Sets for Spherical Radon Transform | p. 339 |
| Spherical Radon transform | p. 339 |
| Conical injectivity sets | p. 341 |
| Description of the kernel of R with respect to sets with spherical symmetry | p. 344 |
| Two-radii theorems for the spherical Radon transform | p. 351 |
| Injectivity sets of the transform R on compactly supported functions | p. 355 |
| Some Questions of Approximation Theory | p. 359 |
| L[superscript p] analogues of the Wiener-Tauberian theorem on subsets of the real line | p. 359 |
| Multidimensional L[superscript p] analogues of the Wiener-Tauberian theorem | p. 361 |
| Approximation by spherical waves | p. 364 |
| Gap Theorems | p. 366 |
| Uniqueness theorem for sequence of exponential polynomials | p. 366 |
| Local theorems for generalized trigonometrical series | p. 371 |
| Gap theorems for the sequences of linear combinations of the cylindrical functions | p. 375 |
| Morera Type Theorems | p. 378 |
| An extremal problem related to Morera's theorem | p. 378 |
| Zalcman's problem and related questions | p. 382 |
| Morera type theorems in domains with the weak cone condition | p. 387 |
| Refinement of Dzyadyk's theorem | p. 388 |
| Mean Value Characterization of Various Classes of Functions | p. 390 |
| Mean value property for harmonic functions | p. 390 |
| Theorems on weighted spherical means for some differential equations | p. 399 |
| Mean value theorems for polyanalytic functions | p. 402 |
| Mean value theorem for harmonic polynomials | p. 404 |
| Applications to Partial Differential Equations | p. 408 |
| Uniqueness theorems for the Darboux equation | p. 408 |
| Uniqueness theorems for the wave equation | p. 409 |
| Stationary sets for the heat equation | p. 414 |
| Some Questions of Measure Theory | p. 416 |
| Measure preserving transformations | p. 416 |
| Measures are determined by their values on balls | p. 418 |
| Functions with Zero Integrals in Problems of the Discrete Geometry | p. 420 |
| Some general estimates for densities of packings | p. 420 |
| Packings by l-admissible sets | p. 423 |
| Inscribed sets | p. 425 |
| Comments and Open Problems | p. 427 |
| Bibliography | p. 430 |
| Author Index | p. 449 |
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