Equiconvergence theorems for Chébli-Trimèche hypergroups
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 2, p. 211-265

We consider a Sturm-Liouville operator of the kind d 2 dt 2 +A ' t Atd dt on 0,+ and the related eigenfunction expansion. We prove that, under suitable assumptions on At, 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 Atdt functions. Our methods rely on asymptotic expansions for the eigenfunctions and the Harish-Chandra function that we prove under very weak hypotheses.

Classification:  43A62,  43A32,  34L10
@article{ASNSP_2009_5_8_2_211_0,
     author = {Brandolini, Luca and Gigante, Giacomo},
     title = {Equiconvergence theorems for Ch\'ebli-Trim\`eche hypergroups},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 8},
     number = {2},
     year = {2009},
     pages = {211-265},
     zbl = {1171.43005},
     mrnumber = {2548246},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2009_5_8_2_211_0}
}
Brandolini, Luca; Gigante, Giacomo. Equiconvergence theorems for Chébli-Trimèche hypergroups. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 8 (2009) no. 2, pp. 211-265. http://www.numdam.org/item/ASNSP_2009_5_8_2_211_0/

[1] W. R. Bloom and H. Heyer, “Harmonic Analysis of Probability Measures on Hypergroups”, de Gruyter Studies in Mathematics, Vol. 20, 1994. | MR 1312826

[2] W. R. Bloom and Z. Xu, The Hardy-Littlewood maximal function for Chébli-Trimèche hypergroups, Contemp. Math. 183 (1995), 45–70. | MR 1334771 | Zbl 0834.42011

[3] M. Bôcher, On regular singular points of linear differential equations of the second order whose coefficients are not necessarily analytic, Trans. Amer. Math. Soc. 1 (1990), 40–52. | JFM 31.0345.01

[4] O. Bracco, “Fonction Maximale Associée à des Opérateurs de Sturm-Liouville Singuliers”, Thèse, Université Louis Pasteur, Strasbourg, 1999. | MR 1691449

[5] H. Chebli, Sur un Théorème de Paley-Wiener Associé à la Décomposition Spectrale d’un Opérateur de Sturm-Liouville sur ]0,+[, J. Funct. Anal. 17 (1974), 447–461. | MR 500293 | Zbl 0288.47040

[6] L. Colzani, A. Crespi, G. Travaglini and M. Vignati, Equiconvergence theorems for Fourier-Bessel expansions with applications to the harmonic analysis of radial functions in Euclidean and non-Euclidean spaces, Trans. Amer. Math. Soc. 338 (1993), 43–55. | MR 1108610 | Zbl 0785.42006

[7] A. Fitouhi and M. M. Hamza, A uniform expansion for the eigenfunction of a singular second-order differential operator, SIAM J. Math Anal. 21 (1990), 1619–1632. | MR 1075594 | Zbl 0736.34055

[8] M. Flensted-Jensen and T. Koornwinder, The convolution structure for Jacobi function expansions, Ark. Mat. 11 (1973), 245–262. | MR 340938 | Zbl 0267.42009

[9] L. Gallardo and K. Trimèche, Renewal theorems for Singular Differential Operators, J. Theoret. Probab. 15 (2002), 161–205. | MR 1883207 | Zbl 0992.60029

[10] N. N. Lebedev, “Special Functions and their Applications”, Dover Publications, 1972. | MR 350075

[11] F. W. J. Olver, “Asymptotic and Special function”, AKP Classics, 1997. | MR 1429619

[12] C. Meaney and E. Prestini, Almost everywhere convergence of inverse spherical transforms on noncompact symmetric spaces, J. Funct. Anal. 149 (1997), 277–304. | MR 1472361 | Zbl 0883.43012

[13] C. Meaney and E. Prestini, On almost-everywhere convergence of inverse spherical transforms, Pacific J. Math. 170 (1995), 203–215. | MR 1359977 | Zbl 0857.43006

[14] R. J. Stanton and P. A. Tomas, Expansions for spherical functions on noncompact symmetric spaces, Acta Math. 140 (1978), 251–276. | MR 511124 | Zbl 0411.43014

[15] G. N. Watson, “A treatise on the Theory of Bessel Functions”, 2nd edition, Cambridge, University Press, 1996. | MR 1349110

[16] H. Zeuner, One dimensional hypergroup, Adv. Math. 76 (1989), 1–18. | MR 1004484 | Zbl 0677.43003