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. define Then if then 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 |