Equiconvergence theorems for Chébli-Trimèche hypergroups

Brandolini, Luca¹ ; Gigante, Giacomo¹

¹ Dipartimento di Ingegneria dell’Informazione, e Metodi Matematici, Università degli Studi di Bergamo, Viale Marconi, 5, 24044 Dalmine, Italia

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

Résumé

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.

Détail

Classification : 43A62, 43A32, 34L10