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

MSC:
 44A15 Special integral transforms (Legendre, Hilbert, etc.) 44A10 Laplace transform 44A30 Multiple integral transforms
Keywords:
inverse Mellin transforms
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