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I have an infinite sequence (see the graphic) which I want to interpolate with an analytic function. Polynomial interpolation fails due to Runge phenomenon.

What else can I do?

enter image description here

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It depends greatly on your infinite sequence. What is the infinite sequence you have? –  J. M. Jul 30 '11 at 4:26
    
How would you use a polynomial to interpolate an infinite sequence? If you could settle for a mere $C^\infty$ function, you could use a convolution of your function with a bump function. –  Samuel Jul 30 '11 at 4:35
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Whatever else you do, it seems you should start out with an ansatz $f(x)=-g(x)\cos\pi x$ and interpolate $g$. That is if you mean "analytic" in the technical sense of the word -- if you're just looking for an expression in closed form, you could use $a_n=(-1)^{n-1}b_n$ and then interpolate $b_n$. –  joriki Jul 30 '11 at 5:15
    
What about a complex function? I could imagine, that the given points are on some conic with hyperbolic decreasing circumference. (so that a curve spirals around the x-axis using the z-axis as well). If you could provide some (more!) data I could play a bit... –  Gottfried Helms Jul 30 '11 at 8:19
    
With that imagination of a conic I meant the following. Express a sequence of complex values $z_k$ by your y-values (using a sequential index k=1,2,3,... ) as $z_k=abs(y_k)*\exp(i*\pi*x_k) $ and try to find a logarithmic or exponential interpretation for $abs(y_k) $ in terms of k. –  Gottfried Helms Jul 30 '11 at 8:56

1 Answer 1

You can actually get entire functions this way. See Ahlfors 5.2.3 (canonical products) Theorem 7 (Weierstrass) and Exercise 1 (with hint). You get series expansions at least some of which (many) are convergent.

Seems to have finally worked, scan from pages 194 and 196 of Complex Analysis by Lars V. Ahlfors, second edition 1966.

Trying jpeg, Lineart setting, 600 dpi, evidently about 5.1 MB. This is the highest degree of difficulty, the judges have been coming down pretty hard on gymnasts who overreach, but after a disappointing rotation on the vault it is time to throw it all out there for the floor exercise. No, after three minutes attempting to download, she is going with the more conservative 300 dpi. Not a popular choice, there are boos from around the arena. But at 1.4 MB it seems she has just about avoided disqualification, just nine people-seconds to upload on imgur.

enter image description here

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