Can someone explain the fourier series? I can't get my head around this shit

wildberger disapproves

Cool vid, tho. Wish I had that 3d visualization when I was learning about this shit

reminds me of
youtube.com/watch?v=QVuU2YCwHjw

You can either look at it like approximating everything with a sin or a cos function, if you put in the equations into a computer then it will spill out relatively good approximations. You can even do it for any function with the Fourier transformation.
From a mathematical standpoint you choose sin as the base for your function vector space, which has infinite dimensions and you can have infinite linear independant sin functions they are already a base of your vector space, which means you can represent every continious function with infinite sin functions.

> Cool vid, tho. Wish I had that 3d visualization when I was learning about this shit
same
What's special about cosines and sines?
What kind of functions can be used to decompose functions?
Only periodic ones? Any non-transcendental ones?
Do Wildberger's spread, cross, and twist functions work?

OC

...

Any periodical function is an infinite sum of sins
You'll need it in filters

they need to be orthogonal functions in an interval to which your function space can be mapped

with suitable adjustments, spread polynomials are orthogonal, wildberger wrote a paper on this, though he didn't try to compose functions in this way

does it seem to me that there's always going to be a little tip pointing slightly above where the triangle wave starts? doesnt this contradict that you can write this function as an infinite limit of trig functions?

Brilliant explanation

en.wikipedia.org/wiki/Gibbs_phenomenon

fuck i knew it was that, it's still kinda paradoxical that it happens. Fourier series converge uniformly or pointwise to f(x)?

This is probably the simplest explanation.

depends on discontinuities

Can you link the paper?
I've read a couple of his more serious stuff.

I thought I had it saved, but now I can't find it. It might have been one of his students. All I have handy is the mention in "Spread polynomials ... butterfly effect" where he says
>Observe also that the spread polynomials are positive in this range, so do not form an orthogonal family of polynomials in the usual sense unless they are translated vertically.

But I will keep digging because I thought I had that paper and remember liking it.

see also this post
mathoverflow.net/questions/47520/spread-polynomials

I know I once read a paper that expressly went through this, but I can't find it. Please post if you do find it.

Basically, any wave form can be described as a sum of vanilla cos/sin functions. For example, that square wave can be pretty accurately represented by the sum of 5 cosines with varying coefficients. This comes from engineering background though so not sure o=if the mathematical among us have deeper revelations to share.

At the limit, that tip gets infinitesimally thin.

Well you can determine each normalized polynomial, since the norm of each [math]||S_n|| = \sqrt{m_n}[/math] where
[eqn]m_n = \int_0^1 (S_n-\frac{1}{2})^2 = \frac{2n^2-1}{16n^2-4},~n \in [1,\infty)[/eqn] and so you can normalize each spread function rather trivially.

I am not deep into this kind of mathematics, so I don't even understand the derivation of these weighing functions. It seems that just having normalized orthogonal forms of the spread polynomials would be sufficient for a generalized Fourier series.