Does the Mellin transform have a half-iterate ? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Does the Mellin transform have a half-iterate ? (/showthread.php?tid=862) Does the Mellin transform have a half-iterate ? - tommy1729 - 05/06/2014 Does the Mellin transform have a half-iterate ? After all these half-iterates of functions , integrals and derivatives, I wonder ( again ) about half-iterates of integral transforms. regards tommy1729 RE: Does the Mellin transform have a half-iterate ? - JmsNxn - 05/07/2014 Well I posted a reply to this but it deleted it :/ I have looked at this for so long Tommy. It's very closely related to tetration. I'll tell you how I can do it for some functions. $M(f) = \int_0^\infty f(x)x^{s-1}\,dx$ define $\vartheta(w) = \sum_{n=0}^\infty M^n(f)(s) \frac{w^n}{n!}$ Then if $\phi(z) = [\frac{d^z}{dw^z} \vartheta(w)]_{w=0}$ then $\phi(z) = M^z (f)(s)$ However we have to show lots of conditions on convergence and what not. RE: Does the Mellin transform have a half-iterate ? - tommy1729 - 05/07/2014 I notice that Im not so comfortable with many iteration of the Mellin transform. Convergeance issues seem to pop up out of nowhere. For instance if we start with f(x) = exp(x) or f(x) = exp(-x) we get in trouble before we reach the 3rd iteration of the mellin transform. And if we pick an elementary function between exp(x) and exp(-x), I seem to have trouble finding a closed form for every n th mellin transform. Maybe I need to consider using hypergeometrics ? What are standard tables of n th mellin transforms (for s,x > 0) , if that has even been made yet ? regards tommy1729 RE: Does the Mellin transform have a half-iterate ? - tommy1729 - 05/07/2014 Also of intrest to me are integral transforms that are cyclic : Like for almost all f,s and some transform M : M^[3](f)(s) = f(s) but M^[1](f)(s) =/= f(s) and M^[2](f)(s) =/= f(s). I assume this is consistant with Schwartz kernel theorem. As Kernel I assume abelian functions could be used. And maybe some others too ? regards tommy1729 RE: Does the Mellin transform have a half-iterate ? - tommy1729 - 05/07/2014 Ok this has given me an idea for tetration ... The James-tommy method is in progress regards tommy1729