Question

(PDE)(Fourier Series) I'm trying to figure out how to make a fourier expansion given some function and an orthonormal basis; I've gotten both the orthonormal set and the function, but I don't know what I'm supposed to do to proceed, all my book examples are approached in a wholly different manner.

Answer 1

Just, without specific values, say I have an orthonormal basis, \[\hat{g}_0,\ \hat{g}_1,\ \hat{g}_2\]and some random function \[h(x)\]What do I do, based on this information, to make a fourier expansion of h(x)? What conceptually do I need to do to the information I have?

Answer 2

@SithsAndGiggles

Answer 3

the g_i's are are orthonormal basis for h(x) ?

Answer 4

I think so, yeah, but what do I do? (I'm just not familiar with this really at all)

Answer 5

\[x=(x,g_0)g_0+(x,g_1)g_1+(x,g_2)g_2\]
where (a,b) still means inner product
and x is an element of h
--
this is what i have decipher from wikepedia if deciphered correctly
http://en.wikipedia.org/wiki/Orthonormal_basis

Answer 6

"Find the fourier expansion of h(x)=(???) [Given, I just don't want to put it up here] in terms of the orthonormal set (???) [Also given]."

Answer 7

and all of those g's were suppose to have a hat
just forgot to get them one

Answer 8

I understand. And I'm just looking at some sources putting stuff here I think that might be helpful in the problem .

Answer 9

I really wanted to look at my linear algebra book but it doesn't say much about fourier expansions. :(

Answer 10

Anybody got an idea? I'm stumped.

Answer 11

I don't know enough about linear algebra or Fourier analysis I'm afraid :/

Answer 12

Alright, I think I figured it out, but I'm pretty much out of time anyways, lol. The coefficients of a fourier series, a_0, a_(some stuff) b_(some stuff) are all described in terms relative to your original function and your orthonormal basis, I understand how it works for trigonometric orthonormal bases, but I haven't seen it for anything else.

Answer 13

@agent0smith @dan815