• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 My interpolation method [2020] tommy1729 Ultimate Fellow Posts: 1,491 Threads: 355 Joined: Feb 2009 02/20/2020, 01:22 PM (This post was last modified: 02/20/2020, 08:38 PM by tommy1729.) Let g(z) be an interpolation of tetration that is entire. The idea is to “add “ the log branches “ manually “ and at the same time “add “ the functional equation. For instance let for some positive or zero x  $g(z) = \sum_{n=0}^{oo} exp((z-n) exp^{[n]}(x)) exp^{[n]}(x) (-1)^n sin(\pi z) (\pi(z-n))^{-1}$ This is an interpolation of tetration that is entire and fits perfectly at the values exp^[n](x) = g(n). Now let the functional inverse of g(z) be f0(z). This f0 now approximates an slog by interpolation. Notice all this can be generalized to other bases or even other functions or sequences! Now we try to correct this f0. For x> 1 $f_{n+1}(x) = 1+ f_n(ln(x))$ And then take the lim f_n= f_oo = S(z). Now we take the functional inverse of S(z). We call the inverse tet(z). This tet(z) should be tetration. I assume all this converges.  But I would like a formal proof that the lim f_oo exists. This somewhat resembles the 2sinh method or the base change. I assume since we started with entire we end up with C^oo functions at least. We wonder if this is analytic.  And what criterion’s it satisfies ?  I repeat this can be used for other sequences too such as fibonacci like for instance. Or iterations of polynomials. Also other interpolations can be considered.  Variants are welcome. Regards  Tommy1729 tommy1729 Ultimate Fellow Posts: 1,491 Threads: 355 Joined: Feb 2009 02/20/2020, 08:40 PM (This post was last modified: 02/20/2020, 08:41 PM by tommy1729.) Probably  $f_{n+1}(x) = f_n(exp(x))- 1$ Is better. Regards  Tommy1729 « Next Oldest | Next Newest »

 Possibly Related Threads... Thread Author Replies Views Last Post Tommy's Gaussian method. tommy1729 24 3,645 11/11/2021, 12:58 AM Last Post: JmsNxn The Generalized Gaussian Method (GGM) tommy1729 2 319 10/28/2021, 12:07 PM Last Post: tommy1729 Arguments for the beta method not being Kneser's method JmsNxn 54 5,907 10/23/2021, 03:13 AM Last Post: sheldonison tommy's singularity theorem and connection to kneser and gaussian method tommy1729 2 426 09/20/2021, 04:29 AM Last Post: JmsNxn Why the beta-method is non-zero in the upper half plane JmsNxn 0 311 09/01/2021, 01:57 AM Last Post: JmsNxn Improved infinite composition method tommy1729 5 1,199 07/10/2021, 04:07 AM Last Post: JmsNxn A different approach to the base-change method JmsNxn 0 708 03/17/2021, 11:15 PM Last Post: JmsNxn A support for Andy's (P.Walker's) slog-matrix-method Gottfried 4 4,753 03/08/2021, 07:13 PM Last Post: JmsNxn Doubts on the domains of Nixon's method. MphLee 1 987 03/02/2021, 10:43 PM Last Post: JmsNxn Kneser method question tommy1729 9 9,721 02/11/2020, 01:26 AM Last Post: sheldonison

Users browsing this thread: 1 Guest(s)