Is there a Mellin transform or an analogue on $L^2([0,2\pi])$ or $\ell^2(\mathbb{Z})$?

by Marc_Adrien   Last Updated August 13, 2019 18:20 PM

From Wikipedia, the Mellin transform is an isometry $M : L^2(\mathbb{R}^+) \mapsto L^2(\mathbb{R})$,

$$\{M f\} (x) := \frac{1}{\sqrt{2\pi}} \int_{\mathbb{R}^+} x^{-1/2 + \mathrm{i} s} f(x) dx.$$

https://en.wikipedia.org/wiki/Mellin_transform

Does anyone know if there is an analogue of this transformation on $L^2([0,2\pi])$ or $\ell^2(\mathbb{Z})$ ?

Thanks !



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