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I don't think this article is correctly worded. First of all, a continuous function is certainly not uniquely determined by the requirement that it pass through n specified points, nor is this false statement implied by the Stone-Weierstrass theorem. The correct statement is more along the lines of, if you are looking for a periodic interpolating function of the given trigonometric form, then the coefficients are uniquely determined. However, even this is a bit too simplistic because there are multiple possible choices of trigonometric interpolation polynomial due to aliasing. See e.g. the discussion under discrete Fourier transform.
I don't have time to make this article not suck right now, but I thought I should tag it to warn readers, at least. —Steven G. Johnson 23:44, 12 December 2005 (UTC)
So, Gauss derived a FFT? i.e. a n*log(n) FFT? I never knew this. Why all the fuss about Cooley and Tukey then? Was Gauss' result not noticed until after Cooley-Tukey? Lavaka 20:32, 4 March 2007 (UTC)
I believe the formula is not correct. Suppose K = 1, so the polynomial should be in form (remember we are vanishing term) which satisfies and cannot interpolate the following dataset: .
But if we conversely make gone, the problem becomes well-posessed. I believe, the correct version is
This article has not yet been rated on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
I don't think this article is correctly worded. First of all, a continuous function is certainly not uniquely determined by the requirement that it pass through n specified points, nor is this false statement implied by the Stone-Weierstrass theorem. The correct statement is more along the lines of, if you are looking for a periodic interpolating function of the given trigonometric form, then the coefficients are uniquely determined. However, even this is a bit too simplistic because there are multiple possible choices of trigonometric interpolation polynomial due to aliasing. See e.g. the discussion under discrete Fourier transform.
I don't have time to make this article not suck right now, but I thought I should tag it to warn readers, at least. —Steven G. Johnson 23:44, 12 December 2005 (UTC)
So, Gauss derived a FFT? i.e. a n*log(n) FFT? I never knew this. Why all the fuss about Cooley and Tukey then? Was Gauss' result not noticed until after Cooley-Tukey? Lavaka 20:32, 4 March 2007 (UTC)
I believe the formula is not correct. Suppose K = 1, so the polynomial should be in form (remember we are vanishing term) which satisfies and cannot interpolate the following dataset: .
But if we conversely make gone, the problem becomes well-posessed. I believe, the correct version is