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I'm just a lowly software engineer with a M.Sc. in Computing, but I didn't find that first paragraph particularly difficult to understand. The images further support it.

Naturally, to understand it you need to some understanding of the terms used but that's why they each link to their own wikipedia pages.

Am I suffering from knowledge blindness? The Taylor series article on Wikipedia doesn't seem to have a particularly impenetrable introduction. There's a couple phrases here and there that read a little bit awkwardly, but I think all the concepts here are covered in most high school mathematics curricula, except for the very end that briefly alludes to some introductory analysis concepts (the "analytic" portion). (Note that some of the concepts such as convergence here have links to additional articles)

> In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor, who introduced them in 1715.

> If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.

> The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).

https://en.wikipedia.org/wiki/Taylor_series

Perhaps I can understand this because I already spent countless hours across several courses drilling Taylor's series into my head.

The Wikipedia article reads like a summary for someone who already knows what it is. I'd reckon if I hadn't spent years going over the topic, the article would've read like a riddle.

I don't think the introductory paragraph of the Gibbs Phenomenon article is bad. Quite the contrary, I think it's well-written. The opening sentence in the Taylor series article is terrible, though, but it's at least factually correct. True horror occurs when there is a major mistake. Incidentally, I encountered one such mistake just an hour ago. In the Encyclopedia of Mathematics published by Springer, there is an article entitled "Schur determinant lemma" (https://encyclopediaofmath.org/wiki/Schur_determinant_lemma), which says that the determinant of a block matrix (in Matlab's notation) [P Q;R S] is det(PS-RQ). But this in general is wrong unless PR=RP. The result that "Schur determinant lemma" or "Schur determinant formula" refers to is actually det[P Q;R S]=det(P)det(R-SP^{-1}Q) (provided that P is invertible).

Anyone who sat through a basic lecture of Fourier series (and had any experience with basic calc) should be able to parse out every single term in that paragraph and then combine them to understand "truncating a Fourier series at a sharp corner creates ringing"

Good summary. That kind of simple explanation would be useful in many Wikipedia articles.

As a programmer, I have learned the importance of simple, explanatory variables, comments, and abstraction to communicate what is going on without losing the reader in the details.

The issue is that the topic is only relevant to people already familiar with Fourier series and analysis, so the simplified, imprecise reduction isn't useful there.

Wikipedia definitely suffers from inconsistency in who they are writing for.

I was drawn to the article because I work with audio software and thought I might gain some insight into a new technique I might use.

Regardless, even people with deep familiarity in a topic can benefit from succinct explanations.

Sure. As a mathematician, the current line in the article _is_ succinct and also precise. Overkill for an audio engineer (my summary might be better). But if you have a graduate degree in a STEM field the sentence should be interpretable assuming you took/remember analysis 101.

I have no idea who wikipedia wants to serve with its articles since they vary wildly in intended audience.

But presumably Wikipedia is (or at least ought to be) written for people who didn't already sit through any lectures on the subject?

> I have a PhD in machine learning

Do not take that badly, but have you learned actual statistics and the mathematics behind them? Or like the few ML people I've met, do you just use and tweak models?

The part you quoted is extremely clear to me, even without actual pictures: it just means the approximation you use departs from the actual function, tell you where and why.

I can infer there will be an artifact, and how I could try to minimize it.

> Sometimes Gibbs phenomenon leads to an artefact (called Gibbs artefact) that we see in our images of the heart when we are doing a certain sequence called stress perfusion cardiac MRI.

You describe where and when it happens in practice, not why it exists in theory.

Different needs for different people!

I think you have stated it rather bluntly, but I have noticed this from some people - I don't know whether it applies to the commenter or not, so I won't speculate, but there have been times when - while I don't doubt that they are an expert is the study and ethics of machine learning - I noticed a lack of holistic knowledge when it came to (say) the hardware that actually execute that machine learning.

> I think you have stated it rather bluntly,

Oops. I was just talking from experience, where I had some expectations of understanding given the level of expertise professed that weren't met in practice.

I say that and I don't even have a PhD (or any other degree!!) in statistics or ML...

> I noticed a lack of holistic knowledge when it came to (say) the hardware that actually execute that machine learning.

Exactly, there are a lot of people I talk to who seem to use "black boxes", without any understand of how it works.

If they can be more productive, why not? However, when one of their black boxes is broken, they often get stuck: as they can't even understand how it works in normal cases, understanding when it may stop working, or even more importantly, how to fix it then, is just not possible.

I have found that to often be the case with mathematics and statistics.

Sometimes it can be a problem, so I have derived some quick test questions to evaluate the person understanding. They are not Shibboleth: they are not hurtful by virtue of having different "levels" of answer that the person may give.

Typical example for stats/ML: "Can you explain why the LLN happens?" Bonus points if the person just tells me the Normal distribution is defined by a Gaussian that is the Fourier transform of itself.

>"...is the peculiar manner in which the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. The nth partial sum of the Fourier series has large oscillations near the jump, which might increase the maximum of the partial sum above that of the function itself"

I'd say the minimum qualifications of understanding this sentence adequately is:

- knowing the basic terminologies of calculus to understand what "piecewise continuously differentiable" means

- knowing that the Fourier series is some sort of infinite series, which you can get from context if you already know that infinite series is a thing, from Taylor series, which you'd know from calculus from the above requirement.

> introduce the topic in a way the average reader can understand

This is an enormous challenge that is met only by people with a particular gift for communication and teaching... And they are not rewarded for their skill beyond some scattered applause.

In my experience wiki articles on mathematics are really bad for learning about a new topic. What they are good for is brushing up on something you once knew, or learning a few new details about something.

But this is almost a necessity, since people will have so different backgrounds. Just imagine bringing a first-grader up to speed on the gibbs phenomenon. That's a lot of things they need to learn first.

Mathematics is, unlike almost any other subject a tower of learning where everything builds on the previous level. So it's uniquely hard.

To be fair to Wikipedia, Taylor Series are taught to students in the UK at A-level but only for about (thanks to the horrors of A-level mathematics) an A4 page worth of material.

As for the section you quote - I saw what it was trying to say in my head (graphically-ish) before I actually parsed the sentence properly, that might be why it's remained poor. It's not a bad explanation it just isn't long enough.

it's less of an explanation and more of a summary in my opinion.