Equiconvergence theorems for Chébli-Trimèche hypergroups

Brandolini, Luca; Gigante, Giacomo

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 2, pp. 211-265.

We consider a Sturm-Liouville operator of the kind $\frac{d^2}{d{t}^2}+\frac{A^{\text{'}}\left(t\right)}{A\left(t\right)}\frac{d}{dt}$ on $\left(0,+\infty \right)$ and the related eigenfunction expansion. We prove that, under suitable assumptions on $A\left(t\right)$, the partial sums of the Fourier integral associated to such expansion behave like the partial sums of the classical Fourier-Bessel transform. This implies an almost everywhere convergence result for $L^p\left(A\left(t\right)dt\right)$ functions. Our methods rely on asymptotic expansions for the eigenfunctions and the Harish-Chandra function that we prove under very weak hypotheses.

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