Distributional {D}unkl transform and {D}unkl convolution operators

Betancor, Jorge J.

Bollettino dell'Unione Matematica Italiana, Tome 9-A (2006), p. 221-245 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

In this paper, that is divided in two parts, we study the distributional Dunkl transform on R. In the first part we investigate the Dunkl transform and the Dunkl convolution operators on tempered distributions. We prove that the tempered distributions defining Dunkl convolution operators on the Schwartz space ƒ are the elements of 𝒪c′, the space of usual convolution operators on S. In the second part we define the distributional Dunkl transform by employing the kernel method. We introduce Frechet function spaces containing the kernel of the Dunkl transform. In theproof of the properties of the distributional Dunkl transform, defined on the correspoding dual spaces, certain representations of the elements of the dual spaces will play an important role. These representations allows us to simplify, in contrast with the previous and usual methods (see, for instance [7] and [13]), the mentioned proofs. Our new procedure also applies to other distributional integral transforms that had been investigated by other authors (Hankel transforms ([7] and [13]), amongst others).

Résumé

In questo lavoro, che è diviso in due parti, studiamo la trasformata distribuzionale di Dunkl su R. Nella prima parte studiamo la trasformata di Dunkl e gli operatori di convoluzione di Dunkl sulle distribuzioni temperate. Dimostriamo che le distribuzioni temperate che definiscono operatori di convoluzione di Dunkl sullo spazio di Schwartz S sono gli elementi di 𝒪c′ , lo spazio degli operatori convoluzione usuali su S. Nella seconda parte definiamo la trasformata distribuzionale di Dunkl usando il metodo del nucleo. Introduciamo gli spazi funzione di Frechet contenenti il nucleo della trasformata di Dunkl. Nella dimostrazione delle proprieta della trasformata distribuzionale di Dunkl, definita sugli spazi duali corrispondenti, alcune rappresentazioni degli elementi degli spazi duali giocheranno un ruolo importante. Queste rappresentazioni ci permettono di semplificare, in contrasto con i metodi usuali e precedenti (vedi, per esempio [7] e [13]), le sopracitate dimostrazioni. La nostra nuova procedura si applica anche ad altre trasformate distribuzionali integrali che sono state studiate da altri autori (trasformate di Hankel ([7] e [13]), fra le altre).

Détail

Publié le : 2006-02-01

MR 2204909 | Zbl 1179.46035

Bibliographie

[1] Barrios, J.A. - Betancor, J.J., A Kratzel's integral transformation of distributions, Collect. Math., 42 (1) (1991), 11-32. | MR 1181059 | Zbl 0772.46019

[2] Betancor, J.J., Characterizations of Hankel transformable generalized functions, Internat. J. Math. Math. Sci., 14 (2) (1991), 269-274. | MR 1096866 | Zbl 0745.46044

[3] Betancor, J.J. - Marrero, I., The Hankel convolution and the Zemanian spaces $B_\mu$ and $B'_\mu$, Math. Nachr., 160 (1993), 277-298. | MR 1245003 | Zbl 0796.46023

[4] Dunkl, C.F., Differential difference operators associated with reflection groups, Trans. Amer. Math. Soc., 311 (1989), 167-183. | MR 951883 | Zbl 0652.33004

[5] CChettaoui, H. - Trimèche, K., New Paley-Wiener theorems for the Dunkl transform on $\mathbb{R}$, Integral Transforms Spec. Funct., 14 (2) (2003), 97-115. | MR 1969838

[6] Dube, L.S., On finite Hankel transformation of generalized functions, Pacific J. Math., 62 (1976), 365-378. | MR 410365 | Zbl 0329.46044

[7] Dube, L.S. - Pandey, J.N., On the Hankel transform of distributions, Tohoku Math J., 27 (1975), 337-354. | MR 450964 | Zbl 0331.46032

[8] Dunkl, C.F., Hankel transforms associated to finite reflection groups, Contemp. Math., 138 (1992), 123-138. | MR 1199124 | Zbl 0789.33008

[9] Horvath, J., Topological vector spaces and distributions, Vol. 1, (Addison Wesley, 1966). | MR 205028 | Zbl 0143.15101

[10] De Jeu, M.F.E., The Dunkl transform, Invent. Math., 113 (1993), 147-162. | MR 1223227 | Zbl 0789.33007

[11] Jewett, R.I., Spaces with an abstract convolution of measures, Adv. Math., 18(1975), 1-101. | MR 394034 | Zbl 0325.42017

[12] Kingman, J.F.C., Random walks with spherical symmetry, Acta Math., 109 (1965), 11-53. | MR 149567 | Zbl 0121.12803

[13] Koh, E.L. - Zemanian, A.H., The complex Hankel and I-transformations of generalized functions, SIAM J. Appl. Math., 16 (5) (1968), 945-957. | MR 201930 | Zbl 0167.41503

[14] Marrero, I. - Betancor, J.J., Hankel convolution of generalized functions, Rendiconti di Matematica, 15 (1995), 351-380. | MR 1362778 | Zbl 0833.46026

[15] Ben Mohamed, H. - Trimeche, K., Dunkl transforms on $\mathbb{R}$ and convolution product on new spaces of distributions, Integral Transforms Spec. Func. 14 (5) (2003), 437-458. | MR 2006001 | Zbl 1057.46033

[16] Roesler, M., Convolution algebras which are not necessarily probability preserving, In: Applications of hypergroups and related measure algebras (Summer research Conference, Seattle, 1993), Contemp. Math., 183 (1995). | MR 1334785

[17] Roesler, M., Bessel type signed hypergroups on $\mathbb{R}$, In: H Heyer, A. Mukherjea (Eds.), Proc. XI, Probability measures on groups and related structures, Oberwolfach, 1994, World Scientific, Singapore, 292-304. | MR 1414942

[18] Schwartz, L., Theorie des distributions, Hermann, Paris, 1978. | MR 209834 | Zbl 0399.46028

[19] Soltani, F., $L^p$-Fourier multipliers for the real Dunkl operator on the real line, J. Funct. Anal., 209 (1) (2004), 16-35. | MR 2039216 | Zbl 1045.43003

[20] Treves, F., Topological vector spaces, distributions and kernels, Academic Press, New York, 1967.New York, 1967. | Zbl 0171.10402

[21] Watson, G.N., A treatise on the theory of Bessel functions, Cambridge University Press, 1966. | Zbl 0174.36202

[22] Zemanian, A.H., A distributional K-transformation, SIAM J. Appl. Math., 14 (1966), 1350-1365.

[23] Zemanian, A.H., Generalized integral transformations, Interscience publishers,New York, 1968.