How to interpolate a sequence?

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?

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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
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