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This book presents the first systematic and unified treatment of the theory of mean periodic functions on homogeneous spaces. This area has its classical roots in the beginning of the twentieth century and is now a very active research area, having close connections to harmonic analysis, complex analysis, integral geometry, and analysis on symmetric spaces. The main purpose of this book is the study of local aspects of spectral analysis and spectral synthesis on Euclidean spaces, Riemannian symmetric spaces of an arbitrary rank and Heisenberg groups. The subject can be viewed as arising from three classical topics: John's support theorem, Schwartz's fundamental principle, and Delsarte's two-radii theorem. Highly topical, the book contains most of the significant recent results in this area with complete and detailed proofs. In order to make this book accessible to a wide audience, the authors have included an introductory section that develops analysis on symmetric spaces without the use of Lie theory. Challenging open problems are described and explained, and promising new research directions are indicated. Designed for both experts and beginners in the field, the book is rich in methods for a wide variety of problems in many areas of mathematics.

Harmonic analysis. --- Lie groups. --- Nilpotent Lie groups. --- Periodic functions. --- Symmetric spaces. --- Harmonic analysis --- Symmetric spaces --- Nilpotent Lie groups --- Operations Research --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Spectral synthesis (Mathematics) --- Synthesis, Spectral (Mathematics) --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Mathematics. --- Mathematical analysis. --- Analysis (Mathematics). --- Approximation theory. --- Fourier analysis. --- Functional analysis. --- Integral equations. --- Special functions. --- Analysis. --- Functional Analysis. --- Fourier Analysis. --- Integral Equations. --- Special Functions. --- Approximations and Expansions. --- Special functions --- Mathematical analysis --- Equations, Integral --- Functional equations --- Functional analysis --- Functional calculus --- Calculus of variations --- Integral equations --- Analysis, Fourier --- Theory of approximation --- Functions --- Polynomials --- Chebyshev systems --- 517.1 Mathematical analysis --- Math --- Science --- Group theory --- Spectral theory (Mathematics) --- Banach algebras --- Calculus --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Global analysis (Mathematics). --- Functions, special. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic

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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.

Algebra --- Algebraic geometry --- Differential geometry. Global analysis --- Harmonic analysis. Fourier analysis --- Mathematical analysis --- Mathematics --- algebra --- analyse (wiskunde) --- differentiaal geometrie --- Fourierreeksen --- functies (wiskunde) --- mathematische modellen --- wiskunde --- geometrie