6
$\begingroup$

Consider an integral transform of Borel measures supported on $\mathbb{R}^n_+$ given by $$ f(z) =\int\limits_{\mathbb{R}^n_+} x^{z}\frac{\mu(dx)}{x} $$ where $z = (z_1,...,z_n) \in \mathbb{C}^n$, $x^z = x_1^{z_1}...x_n^{z_n}$ and $\frac{1}{x} = \frac{1}{x_1...x_n}$. This transform generalizes the classical Mellin transform. Is there some literature where I can read about it? Is there an inversion theorem for this case?

$\endgroup$

1 Answer 1

1
$\begingroup$

I found the following paper dealing with multidimensional Mellin inversion (it is not open access, though): https://iopscience.iop.org/article/10.1070/RM2007v062n05ABEH004459/pdf. It gives an inversion theorem for suitable classes of functions.

$\endgroup$
2
  • $\begingroup$ Also [Izvestiya Mathematics vol. 198 iss. 4] Antipova, Irina A - Inversion of many-dimensional Mellin transforms and solutions of algebraic equations (2007) [10.1070_SM2007v198n04ABEH003844] $\endgroup$ Mar 24, 2022 at 0:23
  • $\begingroup$ Also "Mellin Transforms and Algebraic Functions" by Antipovaa and Zykova (researchgate.net/publication/…). $\endgroup$ Nov 17, 2022 at 17:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.