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Chapter I Integral Geometry and Radon Transforms
Chapter II Invariant Differential Operators
Chapter III Invariants and Harmonic Polynomials
Chapter IV Spherical Functions and Spherical Transforms
Chapter V Analysis on Compact Symmetric Spaces
analytic arbitrary assume bijection Cartan Chapter commutative compact group complex consider constant converges coordinate Corollary corresponding decomposition deﬁned denote the set differential operator eigenfunction eigenspace element equation Euclidean ﬁnd ﬁnite ﬁnite-dimensional ﬁrst ﬁxed follows Fourier series Fourier transform function f given Haar measure Harish-Chandra harmonic harmonic polynomials Helgason Hence holomorphic homogeneous space hyperplane implies induced integral invariant measure inversion formula irreducible isometry K-invariant Killing form LaplaceóBeltrami operator Laplacian Lemma Let G Lie algebra Lie group linear Math neighborhood nilpotent normal obtain orbit orthogonal polynomial positive deﬁnite Proposition prove radial Radon transform representation of G restriction result Riemannian manifold Riemannian structure roots satisﬁes semisimple Lie sinh sphere spherical function spherical transform subalgebra subgroup submanifold subset subspace Suppose surjective symmetric space Theorem 4.3 tion topology totally geodesic vector ﬁeld vector space Weyl group