Antipova, I. A. Inversion of multidimensional Mellin transforms. (English. Russian original) Zbl 1148.44003 Russ. Math. Surv. 62, No. 5, 977-979 (2007); translation from Usp. Mat. Nauk 62, No. 5, 147-148 (2007). The aim of this paper is to find suitable classes of functions of several variables in which one has both inversion formula \(MM^{-1}=I=M^{-1}M\) for the direct and inverse Mellin transforms \(M\) and \(M^{-1}\). One result is given below:Theorem 1. If \(\Phi(x)\in M^U_{\theta}\) (a vector space), then its Mellin transform is well defined and belongs to \(W^{\theta}_U\) (another vector space of different functions) and the formula \(M^{-1}M[\Phi]= I[\Phi]\) is valid, that is, \[ \frac1{(2\pi i)^n} \int_{a+i\mathbb R^n} x^{-z}dz\int_{\mathbb R^n_+}\Phi(\zeta)\zeta^{(z-1)} d_{\zeta}=\Phi(x). \] Reviewer: R. S. Dahiya (Ames) Cited in 3 Documents MSC: 44A15 Special integral transforms (Legendre, Hilbert, etc.) 44A10 Laplace transform 44A30 Multiple integral transforms Keywords:inverse Mellin transforms PDF BibTeX XML Cite \textit{I. A. Antipova}, Russ. Math. Surv. 62, No. 5, 977--979 (2007; Zbl 1148.44003); translation from Usp. Mat. Nauk 62, No. 5, 147--148 (2007) Full Text: DOI