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The book demonstrates the development of integral geometry on domains of homogeneous spaces since 1990. It covers a wide range of topics, including analysis on multidimensional Euclidean domains and Riemannian symmetric spaces of arbitrary ranks as well as recent work on phase space and the Heisenberg group. The book includes many significant recent results, some of them hitherto unpublished, among which can be pointed out uniqueness theorems for various classes of functions, far-reaching generalizations of the two-radii problem, the modern versions of the Pompeiu problem, and explicit reconstruction formulae in problems of integral geometry. These results are intriguing and useful in various fields of contemporary mathematics. The proofs given are “minimal” in the sense that they involve only those concepts and facts which are indispensable for the essence of the subject. Each chapter provides a historical perspective on the results presented and includes many interesting open problems. Readers will find this book relevant to harmonic analysis on homogeneous spaces, invariant spaces theory, integral transforms on symmetric spaces and the Heisenberg group, integral equations, special functions, and transmutation operators theory.

Inhaltsverzeichnis

Chapter 1. Preliminaries

Abstract
The aim of this chapter is to acquaint the reader with some tools of analysis on Euclidean spaces. We recall the definition and basic properties of quasi-analytic classes, distributions and convolutions. In addition, the necessary information concerning the analytic wave front set and spherical harmonics is presented. We shall also give basic formulas for special functions that will be used many times later. The system of notation in this chapter is preserved throughout the book.
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 2. The Euclidean Case

Abstract
In many questions of integral geometry there arise operators of the following type: $$Af(x)\,=\, \frac{1}{{\sqrt \pi }}\,\int\nolimits_0^x {}\, \frac{{tf(t)}} {{\sqrt {x^2 - t^2 } }}dt,\,x \,> \,0.$$ This is the classical Abel transform, which can be explicitly inverted.
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 3. Symmetric Spaces of the Non-compact Type

Abstract
From a global viewpoint, a symmetric space is a Riemannian manifold which possesses a symmetry about each point, that is, an involutive isometry leaving the point fixed. This generalizes the notion of reflection in a point in ordinary Euclidean geometry. The theory of symmetric spaces implies that such spaces have a transitive group of isometries and can be represented as coset spaces G/K, where G is a connected Lie group with an involutive automorphism G whose fixed point set is (essentially) K.
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 4. Analogies for Compact Two-point Homogeneous Spaces

Abstract
The operators $$\mathfrak{U}_\delta$$ which we studied in Chapter 3 have analogues in the compact case. In this chapter we study their properties for compact two-point homogeneous spaces of dimension > 1. These are the Riemannian manifolds M with the property that for any two pairs of points $$(p_1,\,p_2)\,{\rm and}\,(q_1,q_2)\,{\rm satisfying}\, d((p_1,\,p_2)\,=\,(q_1,q_2)\,{\rm where}\, d{\rm \,is\, the \,distance \,on}\, M$$, there exists an isometry mapping $$(p_1\,to\,p_2)\,{\rm and}\,(p_1\,to\,p_2).$$ By virtue of Wang’s classification (see Helgason [H5, Chapter 1, § 4]) these are also the compact symmetric spaces of rank one. Unlike the non-compact case, the treatment in this chapter is based on the realizations of the spaces under consideration. Accordingly, the use of Lie theory is minimal.
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 5. The Phase Space Associated to the Heisenberg Group

Abstract
Here we give an analog of the theory developed in Chapter 4 for the case of the phase space ℂ n with the twisted convolution $$(f_1\,\star\,f_2)\,=\, \int_{\mathbb{C}^n} f_1 (z - w)f_2 (w)e^{\frac{i} {2}IM\,\left\langle {\left. {z,w} \right\rangle }\mathbb{C} \right.} \,dw.$$
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 1. Functions with Zero Ball Means on Euclidean Space

Abstract
Functions with vanishing integrals over all balls of a fixed radius can be regarded as a generalization to the multidimensional case of periodic functions on the real line.
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 2. Two-radii Theorems in Symmetric Spaces

Abstract
The results in Chapter 1 suggest the general problem of investigating functions with zero ball means on homogeneous spaces. This chapter deals with the case of symmetric spaces with stress on two-point homogeneous spaces. We treat the spaces of the non-compact type in Sections 2.1–2.6 and the compact case in Section 2.7. Interesting analogies and differences appear.
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 3. The Problem of Findinga Function from Its Ball Means

Abstract
As mentioned earlier, under the assumptions of Theorem 1.8(i)–(iv) the problem of determining a function f in B R by means of its integrals over balls B ri (x) (∣x∣ < R-r i , i = 1, 2) has a unique solution. In Section 3.1, we present an inversion procedure due to Berenstein, Gay and Yger [B20].
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 4. Sets with the Pompeiu Property

Abstract
The statement of Proposition 1.1 shows that in the case where A is a ball, the map $$\mathcal{P}\,:\,\mathcal{C}(\mathbb{R}^n)\,\rightarrow\,\mathcal{C}\left(\rm M(n)\right){\rm \,ginen \, by }\,(\mathcal{P}f)(g)\,=\,\int_{g^-1A}f(x)dx$$
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 5. Functions with Zero Integrals over Polytopes

Abstract
In Chapter 4 we discussed the problem about the ball of smallest radius in which a given set A has the Pompeiu property (see the definition of R(A) in Section 4.4).
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 6. Ellipsoidal Means

Abstract
It follows from the Brown–Schreiber–Taylor theorem that an arbitrary ellipsoid R which is different from a ball has the Pompeiu property (see Section 4.1).
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 7. The Pompeiu Property on a Sphere

Abstract
In Section 2.7 we studied the class of functions, vr(B R), having vanishing integrals over all closed balls of radius r lying in B R.
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 8. The Pompeiu Transform on Symmetric Spaces and Groups

Abstract
In this chapter we give a general definition of the Pompeiu transform and study its injectivity sets. In Section 8.1, we present sufficient conditions for injectivity of the Pompeiu transform on symmetric spaces of the non-compact type. In Section 8.2, the problem of the description of the injectivity sets for a broad class of distributions with support on the unit sphere is solved.
Valery V. Volchkov, Vitaly V. Volchkov

Chapter 9. Pompeiu Transforms on Manifolds

Abstract
The Pompeiu transform which we studied for symmetric spaces and groups makes sense for an arbitrary complete Riemannian manifold X.
Valery V. Volchkov, Vitaly V. Volchkov

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