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Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets. (English) Zbl 0870.43004
From the authors’ abstract: We consider the Radon transform \(R_\alpha\), \(\alpha\geq 0\), on the Laguerre hypergroup \(K=[0,+\infty[\times\mathbb{R}\). We characterize a space of infinitely differentiable and rapidly decreasing functions together with their derivatives such that \(R_\alpha\) is a bijection from this space onto itself. We establish an inversion formula and a Plancherel theorem for the operator \(R_\alpha\). Finally, by using the continuous wavelet transform on the Laguerre hypergroup \(K\), we deduce another expression for the inverse \(R_\alpha^{-1}\) of the operator \(R_\alpha\).
Reviewer: N.Bozhinov (Sofia)

MSC:
43A62 Harmonic analysis on hypergroups
44A12 Radon transform
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[1] Bloom, W.R.; Heyer, H., Harmonic analysis of probability measures on hypergroups, De gruyter studies in mathematics, 20, (1994), de Gruyter Berlin/New York
[2] Erdelyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G., Higher transcendental functions, (1953), McGraw-Hill New York · Zbl 0052.29502
[3] Faraut, J.; Harzallah, K., Deux cours d’analyse harmonique, (1984), Birkhäuser Boston
[4] Felix, R., Radon-transformation auf nilpotenten Lie-gruppen, Invent. math., 112, 413-443, (1993) · Zbl 0798.44001
[5] Geller, D.; Stein, E.M., Singular convolution operators on the Heisenberg group, Bull. amer. math. soc., 6, 99-103, (1982) · Zbl 0483.43005
[6] Geller, D.; Stein, E.M., Estimates for singular convolution operators on the Heisenberg group, Math. ann., 267, 1-15, (1984) · Zbl 0537.43005
[7] Gonzalez, F.B., Radon transforms on Grassmann manifolds, J. funct. anal., 71, 339-362, (1987) · Zbl 0676.44004
[8] Helgason, S., The Radon transform, Progress in mathematics, (1980), Birkhäuser Boston · Zbl 0453.43011
[9] Jewett, R.I., Spaces with an abstract convolution of measures, Adv. in math., 18, 1-101, (1975) · Zbl 0325.42017
[10] Koorwinder, T.H., The addition formula for Laguerre polynomials, SIAM J. math. anal., 8, 535-540, (1977)
[11] Ludwig, D., The Radon transform on Euclidean space, Comm. pure appl. math., 23, 49-81, (1966) · Zbl 0134.11305
[12] Nessibi, M.M.; Sifi, M.; Trimèche, K., Continuous wavelet transform and continuous multiscale analysis on Laguerre hypergroup, C. R. math. rep. acad. sci. Canada, 17, 73-78, (1995) · Zbl 0841.43014
[13] Solmon, D.C., Asymptotic formulas for the dual Radon transform and applications, Math. Z., 195, 321-343, (1987) · Zbl 0598.44001
[14] Stempak, K., An algebra associated with the generalized Sublaplacian, Studia math., 88, 245-256, (1988) · Zbl 0672.46025
[15] Stempak, K., Almost everywhere summability of Laguerre series, Studia math., 100, 129-147, (1991) · Zbl 0731.42027
[16] Stempak, K., Mean summability methods for Laguerre series, Trans. amer. math. soc., 322, 671-690, (1990) · Zbl 0713.42024
[17] Strichartz, R.S., L^{P}harmonic analysis and Radon transforms on the Heisenberg group, J. funct. anal., 96, 350-406, (1991) · Zbl 0734.43004
[18] Titchmarsh, E.C., The theory of functions, (1939), Oxford Univ. Press London · Zbl 0022.14602
[19] Trimèche, K., Transmutation operators and Mean-periodic functions associated with differential operators, Math. reports, 4, 1-282, (1988) · Zbl 0875.43001
[20] Watson, G.N., A treatise on the theory of Bessel functions, (1966), Cambridge Univ. Press London/New York · Zbl 0174.36202
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