Analyse harmonique L² de la transformée hypergéométrique de Laplace

Abstract : The hypergeometric functions are special functions associated with root systems. They provide a generalization either of Gauss' hypergeometric function (and more precisely of the Jacobi functions) or of the spherical functions on Riemannian symmetric spaces and pseudo-Riemannian noncompacty causal symmetric spaces. In this thesis, we study the L²-harmonic analysis for the so-called -hypergeometric transform. Our main theorem characterizes (under certain hypothesis on the root systems and their multiplicities) the image, under this transform, of the functions which are of class L² with respect to the canonical measure (a) = , Here denotes the multiplicity of the positive root ?. This theorem generalizes to the above mentioned setting, the classical theorem characterizing as a Hardy space the image of the L²-functions on the positive real half-line under the Laplace transform. Some theorems dealing with series decompositions with resoect to special functions are obtained as application of our main theorem
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Mathieu Bohr. Analyse harmonique L² de la transformée hypergéométrique de Laplace. Mathématiques générales [math.GM]. Université Paul Verlaine - Metz, 2010. Français. ⟨NNT : 2010METZ016S⟩. ⟨tel-01752731⟩

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