• 0 Vote(s) - 0 Average
• 1
• 2
• 3
• 4
• 5
 Interpolating an infinite sequence ? tommy1729 Ultimate Fellow Posts: 1,699 Threads: 373 Joined: Feb 2009 06/12/2022, 03:39 PM Consider an infinite sequence of positive reals :  f(n) := a_1 , a_2 , ... Now we want to interpolate to define f(x) for all real x >= 1. Many things are written about interpolation , extrapolation , curve fitting etc. But they usually deal with a finite sequence or finite interval. And adding data changes the entire interpolation function. But I want a stable interpolation of an infinite sequence. ** So when i get 1 , 4 , 9 , 16 , 25 then the interpolation is trivial. But when I am givin complicated sequences and not functions ( like n^3 or taylors : 2 + 3 n + 0.5 n^4 + ... " for integer imput " ) then this way does not work. Im also not looking for best fit , but an actual match. ** I have issues with fractional derivatives and find contour integrals too hard for this. So I came  up with this : a_i = sum  b_n * (i)_n where (i)_n is a kind of falling factorial. In other words : a_1 = b_1 * 1 = b_1 a_2 = b_1 * 2 + b_2 * 2 * 1 = 2 b_1 + 2 b_2 = 2 a_1 + 2 b_2. a_3 = b_1 * 3  + b_2 * 3 * 2 + b_3 * 3 * 2 * 1 = 3 b_1 + 6 b_2 + 6 b_3. etc a_i = b_1 i + b_2 i (i-1) + b_3 i (i-1)(i-2) + b_4 i (i-1)(i-2)(i-3) + ... Notice how the b_n are solvable when the a_i are given. This reduces to linear algebra. This resembles ideas from newton and lagrange. I want to better understand this ( and use it for tetration ). notice that  1 i + 2 i (i-1) + 4 i (i-1) (i-2) + 8 i(i-1)(i-2)(i-3)  + ... does not converge for non-integer i !! So that is problematic as a solution for interpolation. So this creates questions and problems. should be invert the sequence ( replace a_i by 1/a_i ) in case of divergeance and then after interpolation invert again ? Another question is summability methods and ramanujan master theorem. How do they relate ? And ofcourse this falling factorial interpolation is a taylor series in disguise.  So that requires research too.  In fact where does this converge ? It is clearly not within a radius. And how does this relate to other interpolation methods ?? does n^3 interpolate as x^3 ? does f(n) interpolate to f(x) as a continuum sum ; f(x) = sum_0^x  f(x) - f(x-1) or something like that ? And if not , how do they relate ?? We do have the additive property. Vandermonde matrices are related. This all looks very familiar. I even wonder ; how many interpolation methods are there ? How many are interesting ? And how do they relate to dynamical systems ? Finally i want to write : f(x) = v_1 x/2! + v_2 x(x-1)/4! + v_3 x(x-1)(x-2)/6! + v_4 x(x-1)(x-2)(x-3)/8! + ... which converges for bounded v_n. And thus f(x) is an entire function and a consistant interpolation of "something". As mentioned above , we probably wont be able to interpolate 2^^n directly with such ideas but we could perhaps interpolate 1 / 2^^n with this and then take the multiplicative inverse. But we know our method is linear but not how it related to things like multiplicative inverse , summability methods , continuum sum etc. Maybe this is just my lack of a deep understanding of interpolation or linear algebra. Or my memory is getting old. But right now Im puzzled. One more thing  suppose a_i converges to a constant. Can we then use this interpolation as a fixpoint method for dynamical systems ?? regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,699 Threads: 373 Joined: Feb 2009 06/12/2022, 03:44 PM (This post was last modified: 06/12/2022, 03:45 PM by tommy1729.) oh one more thing. this is clearly related to continued fractions. https://en.wikipedia.org/wiki/Euler%27s_...on_formula regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,699 Threads: 373 Joined: Feb 2009 06/12/2022, 03:57 PM (This post was last modified: 06/12/2022, 04:13 PM by tommy1729.) ofcourse im aware of newton's divided differences and the newton polynomial what is basicly the same idea. And perhaps this is useful : https://math.stackexchange.com/questions...a-sequence But that does not answer all my questions. regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,699 Threads: 373 Joined: Feb 2009 06/12/2022, 04:01 PM i used to make the " infinite degree newton polynomial " for primes and prime twins. But without any useful results. JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 06/12/2022, 09:56 PM Interpolating is actually pretty easy. It's when you ask for a functional equation that it's difficult. Assume $$a_n \to \infty$$ and $$b_n\to\infty$$ and we want to find $$f(b_n) = a_n$$. Define a Weierstrass function $$W(z)$$, such that $$W(b_n) = 0$$. Then define: $$f(z) = W(z) \sum_{n=0}^\infty \frac{a_n}{W'(b_n)(z-b_n)}\\$$ You can choose $$W$$ such that the series converges, and that's pretty much it. This is an exercise in John B Conway's complex analysis, if you're looking for a source. tommy1729 Ultimate Fellow Posts: 1,699 Threads: 373 Joined: Feb 2009 06/12/2022, 11:31 PM (06/12/2022, 09:56 PM)JmsNxn Wrote: Interpolating is actually pretty easy. It's when you ask for a functional equation that it's difficult. Assume $$a_n \to \infty$$ and $$b_n\to\infty$$ and we want to find $$f(b_n) = a_n$$. Define a Weierstrass function $$W(z)$$, such that $$W(b_n) = 0$$. Then define: $$f(z) = W(z) \sum_{n=0}^\infty \frac{a_n}{W'(b_n)(z-b_n)}\\$$ You can choose $$W$$ such that the series converges, and that's pretty much it. This is an exercise in John B Conway's complex analysis, if you're looking for a source. hmm Is sin(x) = W(x) valid ? then we get  $$f(z) = sin(z) \sum_{n=0}^\infty \frac{a_n}{cos(2 \pi n)(z-2 \pi n)}\\$$ I guess I made a mistake there ... JmsNxn Ultimate Fellow Posts: 977 Threads: 114 Joined: Dec 2010 06/13/2022, 08:29 PM (06/12/2022, 11:31 PM)tommy1729 Wrote: (06/12/2022, 09:56 PM)JmsNxn Wrote: Interpolating is actually pretty easy. It's when you ask for a functional equation that it's difficult. Assume $$a_n \to \infty$$ and $$b_n\to\infty$$ and we want to find $$f(b_n) = a_n$$. Define a Weierstrass function $$W(z)$$, such that $$W(b_n) = 0$$. Then define: $$f(z) = W(z) \sum_{n=0}^\infty \frac{a_n}{W'(b_n)(z-b_n)}\\$$ You can choose $$W$$ such that the series converges, and that's pretty much it. This is an exercise in John B Conway's complex analysis, if you're looking for a source. hmm Is sin(x) = W(x) valid ? then we get  $$f(z) = sin(z) \sum_{n=0}^\infty \frac{a_n}{cos(2 \pi n)(z-2 \pi n)}\\$$ I guess I made a mistake there ... You'd need to choose a specific zero function depending on $$a_n$$. Such that we have $$W'(b_n)$$ is large enough to force the series to converge. In your cause, you would need to find a function with zeroes at $$n$$ but has a derivative at $$n$$ which causes the series to converge. For example, use: $$f(z) = A(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi A(n) (z-n)}\\$$ This satisfies: $$f(n) = a_n\\$$ And you can force convergence of the series by letting  $$A(n)$$ be as large as possible. So for example $$A(z) = e^z$$ works. tommy1729 Ultimate Fellow Posts: 1,699 Threads: 373 Joined: Feb 2009 06/13/2022, 10:14 PM (This post was last modified: 06/13/2022, 10:16 PM by tommy1729.) (06/13/2022, 08:29 PM)JmsNxn Wrote: $$f(z) = A(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi A(n) (z-n)}\\$$ This satisfies: $$f(n) = a_n\\$$ And you can force convergence of the series by letting  $$A(n)$$ be as large as possible. So for example $$A(z) = e^z$$ works. How does f(5) = a_5 follow from  $$f(z) = \Exp(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi \Exp(n) (z-n)}\\$$ or  $$f(5) = \Exp(5)\sin(2 \pi 5) \sum_{n=0}^\infty \frac{a_n}{2\pi \Exp(n) (z-n)}\\$$ ?? regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,699 Threads: 373 Joined: Feb 2009 06/13/2022, 10:18 PM (06/13/2022, 10:14 PM)tommy1729 Wrote: (06/13/2022, 08:29 PM)JmsNxn Wrote: $$f(z) = A(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi A(n) (z-n)}\\$$ This satisfies: $$f(n) = a_n\\$$ And you can force convergence of the series by letting  $$A(n)$$ be as large as possible. So for example $$A(z) = e^z$$ works. How does f(5) = a_5 follow from  $$f(z) = \Exp(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi \Exp(n) (z-n)}\\$$ or  $$f(5) = \Exp(5)\sin(2 \pi 5) \sum_{n=0}^\infty \frac{a_n}{2\pi \Exp(n) (z-n)}\\$$ ?? regards tommy1729 not sure why tex fails  slightly better How does f(5) = a_5 follow from  $$f(z) = Exp(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi Exp(n) (z-n)}\\$$ or  $$f(5) = Exp(5)\sin(2 \pi 5) \sum_{n=0}^\infty \frac{a_n}{2\pi Exp(n) (z-n)}\\$$ ?? regards tommy1729 tommy1729 Ultimate Fellow Posts: 1,699 Threads: 373 Joined: Feb 2009 06/13/2022, 10:19 PM (06/13/2022, 10:18 PM)tommy1729 Wrote: (06/13/2022, 10:14 PM)tommy1729 Wrote: (06/13/2022, 08:29 PM)JmsNxn Wrote: $$f(z) = A(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi A(n) (z-n)}\\$$ This satisfies: $$f(n) = a_n\\$$ And you can force convergence of the series by letting  $$A(n)$$ be as large as possible. So for example $$A(z) = e^z$$ works. How does f(5) = a_5 follow from  $$f(z) = \Exp(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi \Exp(n) (z-n)}\\$$ or  $$f(5) = \Exp(5)\sin(2 \pi 5) \sum_{n=0}^\infty \frac{a_n}{2\pi \Exp(n) (z-n)}\\$$ ?? regards tommy1729 not sure why tex fails  slightly better How does f(5) = a_5 follow from  $$f(z) = Exp(z)\sin(2 \pi z) \sum_{n=0}^\infty \frac{a_n}{2\pi Exp(n) (z-n)}\\$$ or  $$f(5) = Exp(5)\sin(2 \pi 5) \sum_{n=0}^\infty \frac{a_n}{2\pi Exp(n) (z-n)}\\$$ ?? regards tommy1729 site crashed in my browser ... maybe that relates. tex looked different without changing it. « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post A random question for mathematicians regarding i and the Fibonacci sequence. robo37 1 4,061 06/27/2022, 12:06 AM Last Post: Catullus Infinite tetration and superroot of infinitesimal Ivars 129 214,518 06/18/2022, 11:56 PM Last Post: Catullus Improved infinite composition method tommy1729 5 2,555 07/10/2021, 04:07 AM Last Post: JmsNxn [repost] A nowhere analytic infinite sum for tetration. tommy1729 0 3,496 03/20/2018, 12:16 AM Last Post: tommy1729 Remark on Gottfried's "problem with an infinite product" power tower variation tommy1729 4 10,265 05/06/2014, 09:47 PM Last Post: tommy1729 applying continuum sum to interpolate any sequence. JmsNxn 1 5,356 08/18/2013, 08:55 PM Last Post: tommy1729 Problem with infinite product of a function: exp(x) = x * f(x)*f(f(x))*... Gottfried 5 13,475 07/17/2013, 09:46 AM Last Post: Gottfried Wonderful new form of infinite series; easy solve tetration JmsNxn 1 7,269 09/06/2012, 02:01 AM Last Post: JmsNxn Infinite tetration of the imaginary unit GFR 40 100,192 06/26/2011, 08:06 AM Last Post: bo198214 find an a_n sequence tommy1729 1 4,635 06/04/2011, 10:10 PM Last Post: tommy1729

Users browsing this thread: 1 Guest(s)