Talk:Whittaker–Shannon interpolation formula

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[edit] Excessive dependent parameters

  1. \frac{1}{T} = f_s = 2W\,     Less is more. E.g., the article could easily do without W\,.
  2. Also, the figure introduces f_N\,, which is never mentioned in the text.
  3. The formula is actually derived (two different ways) at Nyquist–Shannon sampling theorem.
--Bob K 08:44, 22 March 2006 (UTC)


[edit] Incorrect formula

At [Interpolation_as_convolution_sum], the result:

{T \over 2} \mathrm{rect}(fT)\,

should be:

T \mathrm{rect}(fT)\,

--Bob K 18:56, 22 March 2006 (UTC)

[edit] What's it called?

Nobody calls it the Nyquist–Shannon interpolation formula, and Nyquist had nothing to do with it. Since Whittaker published it before Shannon, he should share the credit. People DO actually call it Whittaker's interpolation formula and the Whittaker–Shannon interpolation formula, though just using Shannon is most common. So, I moved it. Dicklyon 04:48, 3 July 2006 (UTC)

References? — Omegatron 06:18, 3 July 2006 (UTC)
At the time, Shannon described this formula as "common knowlegde in the communication art" and specifically mentions it J.M. Whittaker (1935) among other unnamed mathemathicians. One of them should be G. H. Hardy. On the other hand, Nyqyist knew and published about the critical frequency, but never gave this reconstruction formula.
  • J. R. Higgins: Five short stories about the cardinal series, Bulletin of the AMS 12(1985)
Among other things, such as generalizations of the sampling formula and applications to harmonic analysis and approximation theory, Higgins traces the roots of the sampling formula as far back as to Gauss, Cauchy, Poisson and Borel. He explains that the sampling theorem in both aspects of characterization and reconstruction is first formulated by E.T.Whittaker (1915) and first formally proven by G.H. Hardy (1941). He called the space of bandlimited functions "Paley Wiener space". Since Shannon was a student and collaborator of Wiener, he certainly knew about it.
A modern recount of sampling and reconstruction in approximation theory.
--LutzL 12:05, 3 July 2006 (UTC)
As to what people call it, do a search for various terms on [1]. I see 27 "Shannon interpolation formula", and only a few of anything else beyond "interpolation formula". The "tie breaker" for the "also rans" should be who actually published the formula. See references above. As fas as I can tell Nyquist only used a sinc-like formula for the frequency-domain description of a square pulse or sampling aperture. He was concerned about reconstruction an analog waveform, since the problem he was tackling was recovering the discrete code pulses that were input to a bandwidth-limited channel. Dicklyon 16:13, 4 July 2006 (UTC)
Here's another proposal: let's change it to just the Shannon interpolation formula, since it called that in books about 10X more than anything else. I've just been reading up and found an interesting tidbit about E. T. Whittaker's use of it as his "Cardinal function", in Advances in Shannon's Sampling Theory, by Ahmed I Zayed and Zayed I Zayed. They point out that Whittaker did NOT show this formula or function converges to an original f(t), but rather that it converges to a simplest function consistent with the samples. They say "nowhere in his paper did E. T. Whittaker mention that the cardinal function...was equal to f(t)." To put it more bluntly, he did NOT use to "reconstruct" a function from its samples, which is exactly what Shannon did. So, since Whittaker's name is also rare as a prefix to "interpolation formula" or "reconstruction formula", maybe we should move the article again, to just Shannon's interpolation formula. Or, since it says interpolation, not reconstruction, leave Whittaker's name on it? Opinions? Dicklyon 23:06, 4 July 2006 (UTC)

[edit] Too technical?

The section entitled "Stationary Random Processes" is too technical, and should be updated. Definition of (or link to) \ell^p should be included. --Chrismurf 19:58, 3 December 2006 (UTC)

I removed the "technical" tag after adding the requested link and a few words of explanation. More suggestions for things to try to clarify are welcome. Dicklyon 03:08, 4 December 2006 (UTC)
Thanks - definitely better. It's still pretty technical/dense, but such is the field. I'm afraid I'm not qualified/capable of making it any better than it is now (if in fact there's even anything wrong with how it is now). The added link helps.

Chrismurf 04:52, 8 December 2006 (UTC)

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