Talk:Trigonometric series

what's the point of the anti-example?

Thenub314 just added an "example" of the non-Trigonometric series ${\textstyle \sum _{k=2}^{\infty }\sin(k\theta )/\log(k)}$ but I don’t really get understand why. It seems like an off-topic distraction. There are a million possible other things we could add which are not examples of a trigonometric series, but I don’t see how it helps readers. Can you explain what is notable about this example or why you added it, @Thenub314? This article barely describes what is a trigonometric series. –jacobolus (t) 21:44, 21 February 2023 (UTC)

When I was learning about Trigonometric series, I recall reading there is confusion between trigonometric series and Fourier series. That it was important to keep in mind, every Fourier series was a trigonometric series, but the converse is false. While looking over Hardy's book today, I noticed he gave an explicit example that shows this, so I included it.

Just to clarify though, the example IS a trigonometric series, but it is not a Fourier series. Thenub314 (talk) 01:27, 22 February 2023 (UTC)

I see now. Thanks! I added a couple of figures. –jacobolus (t) 03:53, 22 February 2023 (UTC)

What are all those coefficients?

I tried to read the first section, but the letters A, B, and x aren't defined. Presumably all mathematicians know what is meant, but I sure do not. I am not even sure whether A and B are meant to be constants, though I assume they are.

x is traditionally a variable, but here it's a term in an infinite series, and I have no reason to think it varies, so ... what the heck is it?

I guess I'm saying that I find the entire section impossible to understand, and I took two+ years of undergraduate math getting my degree. Presumably an encyclopedia article should be clear to me. IAmNitpicking (talk) 02:06, 21 March 2023 (UTC)

The As and Bs are the coefficients, analogous to the coefficients $\{a_{n}\}$ in the polynomial $a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n}+\cdots$ of an ordinary power series (or polynomial). I would prefer lower case letters, but apparently capital letters are common in some fields. –jacobolus (t) 02:38, 21 March 2023 (UTC)

The word "coefficient" was right there in the next sentence, but I moved it one sentence forward; does that help? Have you ever seen power series before in your studies, e.g. Taylor series? What about trigonometric polynomials? This is basically a combination of those two ideas. –jacobolus (t) 02:43, 21 March 2023 (UTC)