Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Does the Mellin transform have a half-iterate ?
#1
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
Reply
#2
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.
Reply
#3
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
Reply
#4
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
Reply
#5
Ok this has given me an idea for tetration ...

The James-tommy method is in progress Smile

regards

tommy1729
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 208 09/09/2019, 10:55 PM
Last Post: tommy1729
  Approximation to half-iterate by high indexed natural iterates (base on ShlThrb) Gottfried 1 365 09/09/2019, 10:50 PM
Last Post: tommy1729
  Does tetration take the right half plane to itself? JmsNxn 7 5,625 05/16/2017, 08:46 PM
Last Post: JmsNxn
  Half-iteration of x^(n^2) + 1 tommy1729 3 3,806 03/09/2017, 10:02 PM
Last Post: Xorter
  Uniqueness of half-iterate of exp(x) ? tommy1729 14 14,139 01/09/2017, 02:41 AM
Last Post: Gottfried
  [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried 33 36,349 03/27/2015, 11:28 PM
Last Post: tommy1729
  irrational iterate of 2z(1-z) BenStandeven 2 4,389 08/09/2014, 10:16 PM
Last Post: tommy1729
  [entire exp^0.5] The half logaritm. tommy1729 1 2,350 05/11/2014, 06:10 PM
Last Post: tommy1729
  Simple method for half iterate NOT based on a fixpoint. tommy1729 2 3,201 04/30/2013, 09:33 PM
Last Post: tommy1729
  half-iterates of x^2-x+1 Balarka Sen 2 3,986 04/30/2013, 01:14 AM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)