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?
[Math] How to interpolate a sequence
interpolation
interpolation
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?
Best Answer
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.