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Plancherel theorem for spherical functions
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<p>In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Plancherel theorem for spherical functions is an important result in the <a href="/facts/Group_representation/FsuuDagQ">representation theory</a> of <a href="/facts/Semisimple_Lie_group/m8jA3Bcf">semisimple Lie groups</a>, due in its final form to <a href="/facts/Harish-Chandra/f9De1zCJ">Harish-Chandra</a>. It is a natural generalisation in <a href="/facts/Non-commutative_harmonic_analysis/GYKT5q5b">non-commutative harmonic analysis</a> of the <a href="/facts/Plancherel_formula/IlpoAySd">Plancherel formula</a> and <a href="/facts/Fourier_inversion_formula/aM1aMkf0">Fourier inversion formula</a> in the representation theory of the group of real numbers in classical <a href="/facts/Harmonic_analysis/CyG5bIPb">harmonic analysis</a> and has a similarly close interconnection with the theory of <a href="/facts/Differential_equation/9MGbE3Ca">differential equations</a>. It is the special case for <a href="/facts/Zonal_spherical_function/myzsFsrU">zonal spherical functions</a> of the general <a href="/facts/Plancherel_theorem/IlpoAySd">Plancherel theorem</a> for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the <a href="/facts/Spectral_theory/8sAEEIMM">eigenfunction expansion</a> of radial functions for the <a href="/facts/Laplacian_operator/kov9u3PT">Laplacian operator</a> on the associated <a href="/facts/Symmetric_space/AWYuHfNp">symmetric space</a> <i>X</i>; it also gives the <a href="/facts/Direct_integral/UerMgqEb">direct integral decomposition</a> into <a href="/facts/Irreducible_representation/Hu5geBgk">irreducible representations</a> of the <a href="/facts/Regular_representation/dBND2lRj">regular representation</a> on <i>L</i>2(<i>X</i>). In the case of <a href="/facts/Hyperbolic_space/kI7tnwdk">hyperbolic space</a>, these expansions were known from <a href="/facts/Spectral_theory_of_ordinary_differential_equations/nFN3JjlB">prior results</a> of <a href="/facts/Gustav_Ferdinand_Mehler/MRXaDdqT">Mehler</a>, <a href="/facts/Weyl/4C5HhyGU">Weyl</a> and <a href="/facts/Vladimir_Fock/IeLmHFxd">Fock</a>. </p><p>The main reference for almost all this material is the encyclopedic text of Helgason (1984). </p>