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§ Complex-valued functions is broken.

WikiLink Fourier series in article Discrete-time_Fourier_transform is broken.

Bob K (talk) 19:21, 18 February 2023 (UTC)[reply]

It seems that Eq. (4) should not have /P in exponent. 157.82.122.40 (talk) 07:42, 20 February 2023 (UTC)[reply]

There was a mistake, but it was in the definition of ${\displaystyle c_{n))$, it is necessary to divide by P when reconstructing the function so that the complex exponential have the correct frequency. To see this consider first exponential that multiplies ${\displaystyle c_{1))$, it should have a period of ${\displaystyle P}$ because we are reconstructing a function whose period is ${\displaystyle P}$. Checking that we have

${\displaystyle e^{\frac {2\pi i(x+P)}{P))=e^((\frac {2\pi ix}{P))+{\frac {2\pi iP}{P))}=e^{\frac {2\pi ix}{P))e^{2\pi i}=e^{\frac {2\pi ix}{P))\cdot 1}$

Without dividing by ${\displaystyle P}$ the resulting sums would not be ${\displaystyle P}$ periodic. Thenub314 (talk) 15:00, 20 February 2023 (UTC)[reply]

I am restoring basically the 15-Feb version with some improvements. The recent edits warrant discussion and hopefully a consensus. Bob K (talk) 16:07, 20 February 2023 (UTC)[reply]

OK, I undid the undo. There are two problems I have with the Feb 15th version. First, it incorrectly defines the Fourier series, instead giving formulas for the partial sum. This is (IMO) a serious issue, because there is a lot to talk about in terms of which partial sums you should take and why, which cannot be discussed if you define the series immediately to be the partial sum. The second issue is that very early on in the article into a discussion of the cross correlation function, which was causing some confusion, so I tried to move it. Finally, the revert undid other peoples edits that had nothing to do with the change I introduced, which are presumably the objectionable ones. I will read through it again and try to fix any errors or inconsistencies I see, but I do think this is a better starting point. Thenub314 (talk) 17:22, 20 February 2023 (UTC)[reply]

First of all, I am not committed to the partial sum representation. That is someone else's contribution that I have tried to respectively maintain, because it seems harmless enough (to me). Secondly, I appreciate that you merely "moved" the cross-correlation stuff. And that probably would have been fine, except for the other problems that resulted. Thirdly, yes I realize I undid some other minor edits, but I think they actually do have to do with your changes. Whether or not that is strictly true, they are very minor ones that could easily be redone... not justification for a complete undo.

While you are "reading through it again", consider:

• As you no doubt know, the Fourier series is not limited to periodic s(x). One cycle of it can represent a function in just an interval (or with compact support, as it is sometimes described). Your restriction to periodic s(x) is unnecessary, and inconsistent with Fig. 1.
• The section Complex-valued functions is deleted, and not properly replaced with anything explicit.
• Do you seriously believe that a consensus of editors will prefer the amplitude of component ${\displaystyle n,}$ in terms of its subcomponents, to be ${\displaystyle a_{n}={\sqrt {A_{n}^{2}+B_{n}^{2))))$ instead of ${\displaystyle A_{n}={\sqrt {a_{n}^{2}+b_{n}^{2))))$ ?

There's more, but you'll presumably catch some of them in your read-through.
--Bob K (talk) 23:52, 20 February 2023 (UTC)[reply]

This sentence doesn't make any sense as-is. Unfortunately I don't really understand what it could mean. I am going to remove it for the moment. pglpm (talk) 10:00, 23 February 2023 (UTC)[reply]

Oh, I must of clobbered that sentence trying to edit it, thanks Thenub314 (talk) 13:26, 23 February 2023 (UTC)[reply]

It seems to me that the Hilbert space being considered must be ${\displaystyle L^{2}(-\pi ,\pi )}$, since otherwise we need ${\displaystyle f(-\pi )=f(\pi )}$ by periodicity. jajaperson (talk) 05:25, 6 March 2024 (UTC)[reply]

The redirect Hilbert Spaces and Fourier analysis has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 March 28 § Hilbert Spaces and Fourier analysis until a consensus is reached. 1234qwer1234qwer4 01:32, 28 March 2024 (UTC)[reply]