Introduction to radon transforms : with elements of fractional calculus and harmonic analysis / Boris Rubin.

Includes bibliographical references and indexes.

1. Preliminaries;
2. Fractional integration: functions of one variable;
3. Riesz potentials;
4. The Radon transform on Rn;
5. Operators of integral geometry on the unit sphere;
6. Operators of integral geometry in the hyperbolic space;
7. Spherical mean Radon transforms.

The Radon transform represents a function on a manifold by its integrals over certain submanifolds. Integral transformations of this kind have a wide range of applications in modern analysis, integral and convex geometry, medical imaging, and many other areas. Reconstruction of functions from their Radon transforms requires tools from harmonic analysis and fractional differentiation. This comprehensive introduction contains a thorough exploration of Radon transforms and related operators when the basic manifolds are the real Euclidean space, the unit sphere, and the real hyperbolic space. Radon-like transforms are discussed not only on smooth functions but also in the general context of Lebesgue spaces. Applications, open problems, and recent results are also included. The book will be useful for researchers in integral geometry, harmonic analysis, and related branches of mathematics, including applications. The text contains many examples and detailed proofs, making it accessible to graduate students and advanced undergraduates.