By Sigurdur Helgason

ISBN-10: 0821826735

ISBN-13: 9780821826737

This quantity, the second one of Helgason's awesome 3 books on Lie teams and the geometry and research of symmetric areas, is an creation to group-theoretic tools in research on areas with a bunch motion. the 1st bankruptcy offers with the 3 two-dimensional areas of continuing curvature, requiring purely effortless equipment and no Lie thought. it truly is remarkably available and will be compatible for a first-year graduate path. the rest of the ebook covers extra complex subject matters, together with the paintings of Harish-Chandra and others, yet specially that of Helgason himself. certainly, the exposition will be obvious as an account of the author's great contributions to the topic. bankruptcy I offers with smooth essential geometry and Radon transforms. the second one bankruptcy examines the interconnection among Lie teams and differential operators. bankruptcy IV develops the speculation of round capabilities on semisimple Lie teams with a undeniable measure of completeness, together with a research of Harish-Chandra's $c$-function. The remedy of research on compact symmetric areas (Chapter V) contains a few finite-dimensional illustration conception for compact Lie teams and Fourier research on compact teams. every one bankruptcy ends with routines (with recommendations given on the finish of the book!) and ancient notes. This ebook, that is new to the AMS publishing application, is a superb instance of the author's famous transparent and cautious writing sort. It has turn into the traditional textual content for the research of round services and invariant differential operators on symmetric areas. Sigurdur Helgason was once provided the Steele Prize for teams and Geometric research and the better half quantity, Differential Geometry, Lie teams and Symmetric areas.

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**Sample text**

Proof. Let T e £ ' have the Fourier series (7). Then for each cu b > 0 and each i: > 0, T has a continuous restriction to Ea + r0 and to £ 0 >b + r . 8 [Intro. (z)| z |z|>K M < * < sup \fN(z)\ |z|

9sn) denote the Cartesian coordinates of points on S" l. 3. 1. (i) The eigenspaces ofL on S"~l are of the form Ek = span of {famk(s) = (alsi + ••• + ansn)k, seSn~1}. , an)eCn is an isotropic vector and keZ+. The eigenvalue is — k(k + n — 2). (ii) Each eigenspace representation is irreducible. (iii) L2(Sn~l) = YiO Ek (orthogonal Hilbert space decomposition). Proof. Let Pk denote the space of homogeneous polynomial functions p(x 1 ? , x„) of degree k on Rn and Hk cz Pk the subspace of harmonic polynomials.

26). It should be observed that because of (8) the Euclidean and the nonEuclidean harmonic functions coincide. 1. In order to describe the solution to Problem B we need the concept of an analytic functional {hyperfunction) on a compact analytic manifold. Let s/(B) denote the space of analytic functions on the boundary B, considered as an analytic manifold. For each such manifold B, stf(B) carries a certain natural topology; in the case of the circle it can be described rather simply as follows.

### Groups and geometric analysis : integral geometry, invariant differential operators, and spherical functions by Sigurdur Helgason

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