A study of harmonic analysis in orthogonal expansions

我们得到了Hermite展开式、Laguerre展开式和Mehler变换的Hardy型不等式，它们是Hardy空间上经典Hardy不等式的类比。对于Hermite和Laguerre展开，不等式适用于可积函数空间。同时，得到了汉克尔变换的Paley型不等式。此外，我们还证明了具有正拉盖尔系数的函数具有原点处的局部平方可积性意味着全局平方可积性的性质。

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.