Talk:Fourier series

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[edit] Possible Error

In section 5.3, the one on convergence, the following setence appears:

"This is convergence in the norm of the space L2, which means that the series converges almost everywhere to f."

This doesn't seem correct.


[edit] Format of equations

I find the equations difficult to read in their current format. Having a sentence, then a TeX png, then another sentence, etc, makes it hard for me to see what parts are associated with which equation, and where those parts end. Not only that, it makes the section unnecessarily long.

The format that I find very useful is this:

equation \
where
  • some variable is this,
  • some other variable is that, and
  • some variable = this other stuff is this other thing.

Some people might find that ugly or whatever, but it makes it very easy to see what is associated with a single equation. Let me give my example in the format currently employed on this page:

The format that I don't find very useful is this:

equation \

where

some variable

is this,

some other variable

is that, and

some variable = this other stuff

is this other thing.

I find it especially confusing on this page, where more than one *separate* equation is written in one long string. I'm not saying we have to use my perferred format, but I do think that the format needs to be different to make it easier to read. Comments? Fresheneesz 17:41, 24 May 2006 (UTC)

I like more the formal format where everything is indented once. I don't think that the "staircasing" of formulas improves things that much, and is also nonstandard. I'd say we should be conservative and not invent new paradigms here. Oleg Alexandrov (talk) 01:04, 25 May 2006 (UTC)
The bulleting of variables in an equation is not a new paradigm, and I think it applies very nicely here. I started the staircasing of formulas, and so I won't push that if you don't like it. However, I disagree with keeping it the way it is now. If my format is "staircasing" then the current format is "laddering" - the variables, formulas, equations, and explanation, being strung out in a long, hard to read list.
I wonder though, why you removed the bullets. I kept the same indenting, but I feel that the bullets help a reader more clearly see what goes with what. I can't accept the current format, but I would like to hear whatever ideas you have to make those equations more readible.
Do you think its not useful for a reader to be able to understand the equations in a section without reading that whole section? Fresheneesz 02:51, 25 May 2006 (UTC)
I believe the equations are readable enough with just one indent, that's what used in math papers everywhere and people don't complain. :) However, if you do really positively want a star in front of a few equation for emphasis, well, it would not make sense for me to oppose that. Just not much staircasing. :) Oleg Alexandrov (talk) 15:12, 25 May 2006 (UTC)
Alright thanks! (although.. where would people complain about the format of math papers..?) Fresheneesz 04:18, 26 May 2006 (UTC)
I think the last equation (in the "Modern derivations of Fourier series") is wrong; if f is a Riemann-integrable function then the Lebesgue integral (left side) equals the Riemann integral (right side), there's no need to multiply by a constant (2pi in this case).

[edit] relationship between real and complex forms

I was thinking about setting the real and complex forms equal, rather than writing them separately. I would guess people would think this way is "crowded" or "ugly". But I also have some related things I'd like to put up:

Some relationships between the variables in fourier series:

c_n = \frac{a_n - j b_n}{2}
c0 = a0 / 2
an = cn + c − n = 2Re[cn]
bn = c − n − cn = 2Im[cn]

and for real functions:

c_{-n} = c_n^*

Fresheneesz 21:20, 26 May 2006 (UTC)

[edit] Still very convoluted

I'm not new to maths, and I find this article difficult to understand. Possibly it needs to be re-written in a more coherent fashion? Starting with a global definition, f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty [a_n cos( \frac{n \pi x}{L}) + b_n sin( \frac{n \pi x}{L})]
where
a_n = \frac{1}{L} \int_{-L}^{L} f(x) cos(\frac {2 n \pi x}{L}) dx
b_n = \frac{1}{L} \int_{-L}^{L} f(x) sin(\frac {2 n \pi x}{L}) dx

Should the function be odd (link to odd function definition) the fourier cosine series may be used. This simplifies to
a_n = \frac {2}{L} \int_{L_1}^{L_2} f(x) cos(\frac {2 n \pi x}{L}) dx
bn = 0

Should the function be even (link to even function definition) the fourier sine series may be used. This simplifies to
an = 0
b_n = \frac {2}{L} \int_{L_1}^{L_2} f(x) sin(\frac {2 n \pi x}{L}) dx

Then explain the wave (string) equation, complex, real, properties, then historical.

If I have made mathematical mistakes, please forgive me, because I have apparently conflicting sources. :) I am only beginning fourier analysis, so I may have missed formulae which are important to more advanced parts.

Edit: oops, everything *is* in there - I must have skimmed past it these last 10 times I viewed the page. However, I still find the omega, t and T notation confusing - wouldn't f(x) be easier? Chrislewis.au

f(t) is used because in many (or most) cases, fourier series are implimented as functions of time. But you're right that f(x) might imply a more general use - however I think its a very minor thing, and might confuse people used to seeing f(t). Also, the use of t and T as period and time don't have an easily understood analog in other sets - like distance rather than time. So i really don't think that should be changed. I did make a note about what t1 and t2 are explicitely, rather than just compared to the period T. Fresheneesz 19:08, 2 July 2006 (UTC)

Chrislewis, please notice the difference between these:

f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty [a_n cos( \frac{n \pi x}{L}) + b_n sin( \frac{n \pi x}{L})]
f(x) = \frac{a_0}{2} + \sum_{n=1}^\infty \left[a_n \cos\left( \frac{n \pi x}{L}\right) + b_n \sin\left( \frac{n \pi x}{L}\right)\right]
  • The square brackets and round parentheses are bigger in the second one.
  • "sin" and "cos" are not italicized in the second one. The preceeding backslash not only prevents the letters from being italicized as if they were variables, but also provides proper spacing before and after "sin" and "cos" in some circumstances.

Michael Hardy 00:05, 3 July 2006 (UTC)

My apologies - I didnt notice the differences first time. (Chris 07:44, 13 August 2006 (UTC))

I think the original poster's idea is good because now it's really unclear why you can integrate on half of the function if it's odd or even and multiply this by 2. As it is now, there are no explanations and it creates confusion even with the editors. I'll try to come up with something. (LovaAndriamanjay 03:00, 28 November 2006 (UTC))

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