A study of harmonic analysis in orthogonal expansions

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We have obtained Hardy type inequalities for the Hermite expansion, the Laguerre expansions and the Mehler transforms, which are analogues of the classical Hardy inequality on the Hardy space. For Hermite and Laguerre expansions, the inequalities hold on the space of integrable functions. Also, the Paley type inequality for the Hankel transform has been obtained. Moreover, we have showed that the functions with positive Laguerre coefficients have the property that local square integrability at the origin implies global square integrability.