Mourou, M. A.; Trimèche, K. Calderón’s formula associated with a differential operator on \((0,\infty)\) and inversion of the generalized Abel transform. (English) Zbl 0912.42016 J. Fourier Anal. Appl. 4, No. 2, 229-245 (1998). This is a continuation of a series of papers by K. Trimèche and coauthors devoted to continuous wavelet transforms generated by the generalized shift and associated with singular differential operators on the half-line \([0,\infty)\). For a large class of such operators of the Darboux type, having the form \[ \Delta= {d^2\over dx^2}+ {A'(x)\over A(x)} {d\over dx}, \] the authors define the notion of a generalized shift and prove the Calderón-type reproducing formula for the associated wavelet transforms in the relevant \(L^2\)-setting. These transforms are applied to explicit inversion of the corresponding generalized fractional integrals of the Liouville type.Remark: The wavelet transforms technique for the usual fractional integrals of different types was developed by the reviewer in his book “Fractional integrals and potentials” (1996; Zbl 0864.26004). Reviewer: B.Rubin (Jerusalem) Cited in 2 Documents MSC: 42C15 General harmonic expansions, frames 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 44A15 Special integral transforms (Legendre, Hilbert, etc.) Keywords:Abel transform; continuous wavelet transforms; generalized shift; Calderón-type reproducing formula Citations:Zbl 0864.26004 PDFBibTeX XMLCite \textit{M. A. Mourou} and \textit{K. Trimèche}, J. Fourier Anal. Appl. 4, No. 2, 229--245 (1998; Zbl 0912.42016) Full Text: DOI EuDML References: [1] Xu, Z. (1995). The Hardy-Littlewood maximal function for Chébli-Trimèche Hypergroups,Contemporary Math.,183, 45–70. · Zbl 0834.42011 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.