Gibbs Phenomenon

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.

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.

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

> 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.

>"...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.

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"

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).

> 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!

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.

This video taught me more about digital audio in 20 minutes than days of reading Wikipedia would do. I recommend it to anyone even vaguely interested in the subject.

I think this is only superficially similar but actually very distinct. In countersteering, no oscillation is required, even for the most sudden direction changes, only a single movement in the opposite direction is all it takes.