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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,
\[ \frac1{(2\pi i)^n} \int_{a+i\mathbb R^n} x^{-z}dz\int_{\mathbb R^n_+}\Phi(\zeta)\zeta^{(z-1)} d_{\zeta}=\Phi(x). \]

44A15 Special integral transforms (Legendre, Hilbert, etc.)
44A10 Laplace transform
44A30 Multiple integral transforms
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