Checking Fourier Series Quickly - Best Way?

[broegger]

What is the best way to (quickly) check if a calculated Fourier series is the correct one?

 

[matt grime]

look it up on the internet. wolfram has i believe many such series for examination. there is no way to quickly check that any series is correct, and if you want I can give a very complicated explanation of that. The quick one runs: there are infinitely many functions that have the same series, how would you like to distinguish between them? But that probably requires you to know a lot more measure theory.

 

[broegger]

I'm only interested in the cases where the series _do_ converge. Now, a converging Fourier series corresponds to one and only one function (the one that it converges to), am I right (I'm probably not, I know :)?

My problem is this: Given a function and a series (which I calculate), how can I check that this series is the right one KNOWING that the series I am supposed to find DOES converge (uniform and/or pointwise). That is: what is the easiest way to check that a series converges against a given function?

Please remember that this is an introductory course...

 

[matt grime]

A function is not equal to its Fourier series even if they do converge. This was a mistake even Fourier made. Any course that let's you think they are equal needs some revision.

The best way to make sure you have the correct series is not to make mistakes working out the coefficients, beyond that are you asking for something that I don't think exists since it would require you to know how to sum complicated things. You can make qualitative checks: does it have the right zeroes, if it's an even function is it only cos, odd only sin, is it correct when 'x=0'

or you can use the resources on the web and in your library, there are long lists of series that people have worked out, and there are certain ones that get frequently asked such as the saw tooth wave and the square wave, go find them and check there. i'd suggest wolfram, unless someone else has a definite link.

there may even be some computer packages that spew out Fourier series, but I've never had to use any (not cos I'm fantastic at Fourier series, but because I've never had to work out any difficult ones.)

 

[broegger]

Even functions should only involve cosine and odd functions only sine - that's obvious, but I never thought of it, thanks (great, since we're often dealing with even or odd functions).

Do you know any good books that contain a list of important Fourier series. I've heard of something called Schaums Mathematical Tables or something, but I don't know if it's has any Fourier series in it (I know that many series can be found at mathworld.com, but I would like it in book-form for use in the written exam)...

Thanks for taking the time.

 

[matt grime]

If in doubt google for the words you're interested in and add in wolfram. this gives:

http://mathworld.wolfram.com/FourierSeries.html

about 2/3rds of the way down is a table with three series in and a reference to books that give more.

 

[ReyChiquito]

A function is not equal to its Fourier series even if they do converge. This was a mistake even Fourier made. Any course that let's you think they are equal needs some revision.

How so?

I don't understand.

 

[matt grime]

Let's do it T^1 to R.

The function that is identically zero has the Fourier series all of whose coefficients are zero. As does any function which is zero except on a set of measure zero. Thus there are an uncountable number of functions that all have the same Fourier series and are pairwise distinct. It is a simple consequence of the fact that

$$\int_{\mathbb{T}}|f|dx = 0$$

does not imply that f is identically zero.

 

[ReyChiquito]

But that's only under special circumstances, right?

 

[matt grime]

What do you mean special? If you check you'll find that the kinds of functions you think are not special are in fact very special: continuous ones. Mathematically speaking these are not necessarily the norm in l^2, if you'll forgive the pun.

 

[ReyChiquito]

I studied physics, that's what i mean ;) .

In physics we start all problems with "let f be a well behaved function..."

Fourier series (generalized, not only trig) are very important for most physics problems, so when you said "a function is not equal to its Fourier series even if they do converge" you scared me.

I should have know you were talking about some mathematical aberration (lol, no offence).

 

[matt grime]

For more information you should study the difference between

L^2 and l^2, one is the space of lesbegue square measurable functions on T, and the other is the set of square summable series. There is a link between them.