• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Alternative manners of expressing Kneser JmsNxn Long Time Fellow Posts: 571 Threads: 95 Joined: Dec 2010 03/19/2021, 01:02 AM (This post was last modified: 03/19/2021, 01:08 AM by JmsNxn.) I think I may be on the way to answering this question. I've always wondered if Hilbert spaces could be useful for Tetration. This may be the first case I've ever seen it viable. I'll just post what I have here, and the rest of my thoughts I will compose in a PDF. This is an attempt at placing Kneser's tetration in a Hardy space, and deriving uniqueness/existence in this manner. Avoiding the Riemann mapping theorem only in technicality. I'll be brief here. Let's call $\mathcal{T}$ the space of holomorphic functions, $ h(z) = \sum_{n} c_n e^{\lambda_n \sqrt{z}}\,\,\text{for}\,\,z \in \mathbb{C}/\mathbb{R}^{-}\\$ For $\Re(\lambda_n) < 0$. Where the sum is infinite or finite; so long as we are guaranteed convergence. We'll define the inner-product as, for $0 < \sigma < \infty$, $ (f,g) = \lim_{T\to\infty} \frac{1}{2 T}\int_{\sigma-iT}^{\sigma + iT}f(z^2)\overline{g(z^2)}\,dz\\ $ This produces a hilbert space where, $ (e^{\lambda\sqrt{z}}, e^{\omega \sqrt{z}}) = A_{\lambda,\omega}\delta_{\Im(\lambda) = \Im(\omega)}\\$ Where if $\lambda = \omega$ we get $\delta = 1$, if $\Im(\lambda) = \Im(\omega)$ we get a finite real number; and otherwise $\delta = 0$. To those of you who may have fiddled with Hilbert Spaces, this is isomorphic to the space of almost periodic functions, if we look at it only in terms of rays for $z,\lambda$, this is precisely the space of almost periodic functions. This is a space, not exactly a familiar kind of Hilbert Space, but a non-separable Hilbert space. Nonetheless, we can now translate from one Hilbert space to the next. If we call, $\frac{d^z}{dw^z}\vartheta(w)|_{w=0} = f(z)$, where $\vartheta$ exists in a different $L^1$ space, specifically, $ \int_0^\infty |\vartheta(x)|x^{\sigma-1}\,dx < \infty\\$ Of which their exists an inner product $[\vartheta,\theta] = (f,g)$ (This can be explained better with a good understanding of Hilbert spaces; this is just a mapping between spaces). In this new space the inner product looks like, $ [\vartheta,\theta] = \int_0^\infty \vartheta(x)\overline{\theta(x)}\,d\mu\\$ For an appropriately chosen measure $\mu$ (where again, it requires some work to justify this, but it's something in this vector). This allows us to no longer talk about $f(z) = \text{tet}_K(i\sqrt{iz})$ as a function on its own, but rather talk about other functions which exist in a different Hilbert space, and approximating the solution in the different Hilbert space. I apologize if I'm being a bit top heavy in my analysis right now. I'm going to start a paper on this, mostly with the goal of finding a different manner of defining uniqueness and expressibility of Kneser's Tetration. Even if I have to use Kneser's tetration to do everything I'm going to do; I think it may be very important to view it as a Hilbert Space, and view the construction in a different Hilbert space. « Next Oldest | Next Newest »

 Messages In This Thread Alternative manners of expressing Kneser - by JmsNxn - 03/18/2021, 02:27 PM RE: Alternative manners of expressing Kneser - by JmsNxn - 03/19/2021, 01:02 AM

 Possibly Related Threads... Thread Author Replies Views Last Post Arguments for the beta method not being Kneser's method JmsNxn 54 9,043 10/23/2021, 03:13 AM Last Post: sheldonison tommy's singularity theorem and connection to kneser and gaussian method tommy1729 2 635 09/20/2021, 04:29 AM Last Post: JmsNxn Generalized Kneser superfunction trick (the iterated limit definition) MphLee 25 9,009 05/26/2021, 11:55 PM Last Post: MphLee Questions about Kneser... JmsNxn 2 1,353 02/16/2021, 12:46 AM Last Post: JmsNxn Kneser method question tommy1729 9 10,292 02/11/2020, 01:26 AM Last Post: sheldonison tetration from alternative fixed point sheldonison 22 53,346 12/24/2019, 06:26 AM Last Post: Daniel [split] Understanding Kneser Riemann method andydude 7 15,433 01/13/2016, 10:58 PM Last Post: sheldonison alternative fixpoints = branches ? tommy1729 0 3,308 10/11/2014, 08:50 AM Last Post: tommy1729 Equations for Kneser sexp algorithm sheldonison 18 45,345 06/21/2011, 01:48 AM Last Post: sheldonison "Kneser"-like mapping also for complex base? mike3 2 9,094 11/07/2010, 01:37 PM Last Post: sheldonison

Users browsing this thread: 1 Guest(s)