• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Does the Mellin transform have a half-iterate ? tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 05/06/2014, 10:25 PM 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 JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 05/07/2014, 03:22 PM 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. tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 05/07/2014, 11:17 PM 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 tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 05/07/2014, 11:42 PM 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 tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 05/07/2014, 11:52 PM Ok this has given me an idea for tetration ... The James-tommy method is in progress regards tommy1729 « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post Half-iterates and periodic stuff , my mod method [2019] tommy1729 0 491 09/09/2019, 10:55 PM Last Post: tommy1729 Approximation to half-iterate by high indexed natural iterates (base on ShlThrb) Gottfried 1 728 09/09/2019, 10:50 PM Last Post: tommy1729 Does tetration take the right half plane to itself? JmsNxn 7 6,710 05/16/2017, 08:46 PM Last Post: JmsNxn Half-iteration of x^(n^2) + 1 tommy1729 3 4,375 03/09/2017, 10:02 PM Last Post: Xorter Uniqueness of half-iterate of exp(x) ? tommy1729 14 15,954 01/09/2017, 02:41 AM Last Post: Gottfried [AIS] (alternating) Iteration series: Half-iterate using the AIS? Gottfried 33 40,672 03/27/2015, 11:28 PM Last Post: tommy1729 irrational iterate of 2z(1-z) BenStandeven 2 4,774 08/09/2014, 10:16 PM Last Post: tommy1729 [entire exp^0.5] The half logaritm. tommy1729 1 2,591 05/11/2014, 06:10 PM Last Post: tommy1729 Simple method for half iterate NOT based on a fixpoint. tommy1729 2 3,524 04/30/2013, 09:33 PM Last Post: tommy1729 half-iterates of x^2-x+1 Balarka Sen 2 4,289 04/30/2013, 01:14 AM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)