Talk:Prolate spheroidal wave function

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Contrary to what the first paragraph claims, the PSWFs are not time-limited. If $$\psi_n$$ were in fact time-limited, the second $$Q_T$$ in the expression $$Q_T P_W Q_T \psi_n = \lambda_n \psi_n$$ would be redundant anyway. Based on the original paper by Slepian and Pollack, and using the notation established in this article, the correct expression seems to be


 * $$ P_W Q_T \psi_n = \lambda_n \psi_n $$

where $$\psi_n$$ are bandlimited. Perhaps somebody with more background in this matter can confirm this. Qopzm (talk) 00:39, 8 July 2010 (UTC)

I agree
I agree with this observation, that the PSWF's are not time limited. In fact, they are defined to be those functions which maximise the power contained the region [-T,T] given the constraint that they have unit power on R. I have constructed explicitly these functions and they do not vanish outside a compact interval. Indeed, it is a general result that a function which vanishes on a compact interval necessarily cannot have a frequency content which vanished on a compact interval. I will work on the revision to this incorrect material. angusprain 18:25, 1 March 2013.

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Order of material: re-written
List of changes:
 * Headings inserted, paragraphs re-ordered in more natural way
 * Application on band-limited functions shifted to the middle
 * In that section, time interval set to $$-T \ldots T$$, as in NIST's Digital Library of Mathematical Functions (DLMF) §30.15 (reference added)
 * The explicit form of the band-limited eigenfunctions given in terms of angular PSWF
 * Correction: the PSWF are not time-limited, but band-limited
 * The convention for prolate coordinates modified as in the corresponding wiki page
 * Dead links to Slepian & el papers replaced by DOI
 * The eigenvalue $$\lambda_{mn}(c)$$ is fixed by the boundary values of the angular function at $$\eta = \pm1$$, and not by the behaviour of the radial function (see DLMF §30.13)
 * References tidied up: include page numbers, bibliographic information: journal_name vol (year) page.

Further improvements: --DieHenkels (talk) 12:54, 20 May 2021 (UTC)
 * Wouldn't it be useful to visualise the prolate functions? The DLMF gives simple cross sections in the coordinates $$\xi$$ and $$\eta$$, but it would be great to show a true spheroid. Needs some 3D plotting.

Oblate spheroidal wave function
The article was apparently drafted by the same team as the prolate functions. Similar changes appear natural, since the two sets of functions are intimately related. --DieHenkels (talk) 12:54, 20 May 2021 (UTC)