Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Hyper operator space
Well I've been muddling this idea around for a while. I have been trying to create a hyper operator space and I recently realized the form of this. I'll start as follows:

If is a hyper operator then, is a hyper operator created by forming left composition. I.e:

for all then

Associate to every function that is a finite product a number as follows:

Where and p_n is the nth prime.

Now hyper operator space is the following:

Now define the inner product as follows:

Where quite clearly (f,f) converges for all elements since the terms decay to zero across x and y faster or just as fast as

We say all the functions are dense in

Orthonormalize them to get such that:

Now we have the advantage of being in a Hilbert space and having an orthonormal basis.

The first operator we have is the transfer operator:

Since this operator is well defined for any element of where

exists such that for all values that [s] returns natural numbers at, this is our solution to hyper operators.

I think the key is to invesetigate the inner product.

Possibly Related Threads...
Thread Author Replies Views Last Post
  Thoughts on hyper-operations of rational but non-integer orders? VSO 2 1,621 09/09/2019, 10:38 PM
Last Post: tommy1729
  Is there a function space for tetration? Chenjesu 0 1,005 06/23/2019, 08:24 PM
Last Post: Chenjesu
  A fundamental flaw of an operator who's super operator is addition JmsNxn 4 8,536 06/23/2019, 08:19 PM
Last Post: Chenjesu
  Tetration and Sign operator tetration101 0 892 05/15/2019, 07:55 PM
Last Post: tetration101
  Extended xor to Tetrion space Xorter 10 9,186 08/18/2018, 02:54 AM
Last Post: 11Keith22
  Hyper-volume by integration Xorter 0 1,916 04/08/2017, 01:52 PM
Last Post: Xorter
  Hyper operators in computability theory JmsNxn 5 5,824 02/15/2017, 10:07 PM
Last Post: MphLee
  Recursive formula generating bounded hyper-operators JmsNxn 0 2,084 01/17/2017, 05:10 AM
Last Post: JmsNxn
  holomorphic binary operators over naturals; generalized hyper operators JmsNxn 15 20,318 08/22/2016, 12:19 AM
Last Post: JmsNxn
  on constructing hyper operations for bases > eta JmsNxn 1 3,539 04/08/2015, 09:18 PM
Last Post: marraco

Users browsing this thread: 1 Guest(s)