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

Question

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

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

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 !

Answer 1

Let $f \in L^2_{loc}( \Bbb{R}_{>0} )$ such that $f(2x) = f(x)$. Then we have the Fourier series $$f =\sum_n C_n(f) x^{2i \pi n/\log(2)} = \sum_n C_n(f) e^{2i \pi n\log_2(x) }, \quad C_n(f) = \frac{1}{\log(2)}\int_1^2 f(x) e^{-2i \pi n\log_2(x) }\frac{dx}{x}$$