Talk:Trigonometric polynomial

Error in formula

It seems to me there's something wrong with this line for the real trig polynomial:

$T_{N}(x)=\sum _{n=0}^{N}a_{n}\cos(nx)+\sum _{n=0}^{N}a_{n}\sin(nx)\qquad (x\in \mathbf {R} )$

Shouldn't the coefficients for cos and sin be different, like $a_{n}\cos(nx)$ and $b_{n}\sin(nx)$? Because, otherwise wouldn't this just be a single sum like this:

$T_{N}(x)=\sum _{n=0}^{N}{a_{n}(\cos(nx)+\sin(nx))}$

Something is definitely wrong here...
Edsanville 20:32, 23 Aug 2004 (UTC)

You were correct, i fixed the definition. My mistake. MathMartin 11:45, 27 Aug 2004 (UTC)

Notational inconsistency

The page uses an upright "i" for i, but inconsistently uses italic e for e. I prefer i for the imaginary unit in most cases, but I didn't change the page because I don't want to step on toes. 165.189.91.148 18:51, 1 June 2006 (UTC)

Confusing Notes.

I am confused what the following two lines are trying to say:

Using the relation

$T(x)=\mathrm {e} ^{\mathrm {i} Nx}t(x)\,\!$

we can construct a bijective mapping between the complex trigonometric polynomials and the real trigonometric polynomials. Thus a trigonometric polynomial can be considered a periodic function on the real line, with period some multiple of 2π, or as a function on the unit circle.

First the relation above, while certainly bijective, does not directly give a bijection between real and complex trigonometric polynomails. Also, the second sentence doesn't need the first, so it shouldn't start with "Thus". Thenub314 (talk) 18:39, 9 September 2008 (UTC)

I'm also confused. The second sentence is probably what is meant, and I guess the first sentence is an explanation which I don't quite understand. Those two lines do need to be rewritten. -- Jitse Niesen (talk) 03:03, 10 September 2008 (UTC)

Real case special case of complex?

I'm trying to see if the two separate definitions given in this article are "compatible". That is, I think we should be able to have any real trigonometric polynomial by setting the right coefficients in the general complex case, but the way the complex polynomials are defined doesn't seem to let me. If I set al the coefficients in front of the "i" in the general definitions, I will be forced to match the coefficients for cos(nx) with those of the real polynomial, leaving the sin terms unmatched. — Preceding unsigned comment added by 190.118.32.209 (talk) 03:48, 25 April 2013 (UTC)

Just passing by, it seems to me that you don't need the i in front of b_n sin(nx) for the complex part. cf Rudin Principles of Mathematical Analysis p 186 — Preceding unsigned comment added by 39.120.161.171 (talk) 04:03, 9 July 2013‎ (UTC)

Small problem with the definition(?)

Isn't it a problem to use

a_n, b_n ... for n <= 0 <= N

(two occurrences) whereas b_0 does not exist? 90.33.60.1 (talk) 12:34, 7 April 2024 (UTC)

Let's just take those out, as they are already implied by the summation limits. –jacobolus (t) 01:14, 14 April 2024 (UTC)