Analytica's Phenomenal Fourier Quest

[Analytica]

Yeah, I see you people out there think: what's this, a quest in this forum?
But you see, it's not just an item quest. No. This is a quest for knowledge. Knowledge about the Fourier Transform, to be specific.

Let me explain. I am in my last year of secundary school. So, next year, I'm going to the university! I'm studying mathematics. That means, I have to write a mathematical article for my thesis. Somewhat unintentionally, I showed interest in the Fourier Transform. so, now it has become the subject of my thesis.

so?

The main problem is... I am completly ignorant concering the Fourier Transform! With ignorant I mean: I understand the idea, the meaning of the whole thing, and I know some applications of it,
wahmbulance BUT I DO NOT KNOW THE MATHS! wahmbulance

Don't give up on me just yet. Please. I'm very quick to learn (derivatives cost me only a week) and I'm willing to work day and night if necessary. stressed

The things I want to obtain with this quest are:

First Goal: Full use of the technique of integrating by the end of September

End Goal: Full knowledge of the Fourier Transform by the end of the year

All next thing I have to learn are to be posted here... all suggestions are needed and thus welcomed!

[Analytica]

hi there!
I'm working hard on getting integrals right! I thank everyone who posted so far for their contributions! They are highly appreciated!

So far I've learned about the way definite integrals work, and the way you can calculate the 'amount' of sin(x) you can have in Fourier Series. I know I might be using wrong terminology, this is because half of my notes are in english and the other half is in dutch. sorry for that!

Hi there! There's again some progress recorded! Me and Fourier Series have beocomequite good friends, actually biggrin Thank you again for all your co,tributions, they've proven to be incredibly useful!

[Analytica]

reserved!

[jestingrabbit]

Have you tried the MIT open courseware? In particular, I would start with

http://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/

I've gotta say though, fourier analysis was something that I did in the second year of my bachelor's degree and you are really going to be pushing it to come to much of an understanding of what is going on given that you don't have much experience with integrals yet.

[viper232]

"Full knowledge" really might be pushing it, I've known grad students who still read the wiki to refresh on that topic. You really do need to start with understanding how integrals work, start delving into differential equations and complex analysis, and refresh any knowledge you might have of sequences and series.

Once you start learning how to apply series to solve partial differential equations, like using a taylor series to solve an equation about a point, and the like, you should be able to start getting into fouier series and fouier transformations. If you want a really decent understanding, or as you say, a "full understanding", you can't just jump into the wiki article and read and learn... you'll need to learn the fundamentals first. You're not about to learn how to solve the heat equation if you've never seen a PDE in your life.

[Gharbad]

It may not be entirely true to the real math but if you want an 'easy' introduction to Fourier analysis you can go to any electronics circuit theory textbook... it's one of the essential tools and engineers are notorious for disliking the nuances of the math you'd get in a real math book.

You're still going to need calculus and some basic complex analysis.

[jestingrabbit]

Gharbad

It may not be entirely true to the real math but if you want an 'easy' introduction to Fourier analysis you can go to any electronics circuit theory textbook... it's one of the essential tools and engineers are notorious for disliking the nuances of the math you'd get in a real math book.

You're still going to need calculus and some basic complex analysis.

This is pretty true, but you still need a pretty solid grounding in calculus before you can take this approach.

You also don't particularly need complex , you can just go for a cos-series and sine-series approach.

viper232

Once you start learning how to apply series to solve partial differential equations,

You also don't need this. That's more eigen functions as a basis for a solution space stuff, which is very useful, but not necessary.

Truth be told, you just need calculus, in particular an ability to integrate things like f(x)cos(nx) and f(x)sin(nx) where n is any natural number (so integration by parts is pretty key) to do basic fourier stuff.

I'd say that things like understanding L^2 spaces, the sense in which the fourier series converges to the function and Gibbs phenomenon are the next step on the journey, which all comes before using it for differential equations.

[Vannak]

Honestly find a math grad or high level undergrad student who's friendly and willing to sit down with you for 6 hours over a week or two and explain the math required. Tutors will have the best experience, but you won't get a full education on the topic, though in your position that may not be possible anyways.

A decent quality coffee brewer and or press would probably go a long ways as a gift.

[Resid3nt]

Oh, Fourier!

My personal demon---

[VorpalNeko]

Understanding the Fourier transform via the Fourier series first is a good approach. That's the f(x)cos(nx), and f(x)sin(nx) integrations referenced to above, since you would need to know enough calculus to do that anyway. It really helps in understanding how Fourier "picks out" the contribution of a particular frequency. It's literally a projection.

A brief roadmap may be something like this:
(1) For the usual vectors in Euclidean 3-space in Cartesian coordinates, the inner product <u|v> = u_1v_1 + u_2v_2 + u_3v_3 is the product of the lengths of those vectors times the cosine of the angle between them: <u|v> = |u|v|cos θ. The length of u is just sqrt[<u|u>], which can be seen from the distance formula. (For complex-valued coordinates, take conjugate of the components of the first vector in the above.) In particular, note that <u|v> = 0 for nonzero vectors means that they're orthogonal (perpendicular) to each other, as then cos θ = 0.
(2) That means we can project a vector onto another. If û has length 1, then drawing û and v from the same origin forms a right triangle with v as the hypotenuse; dropping v onto û has length proportional to the cosine of the angle between them. Thus, the projection of v onto û is just <û|v>û.
(3) In general, if we have any orthonormal (mutually orthogonal and length 1) vectors {e1,e2,e3}, we can write any given v as v = <e1|v>e1 + <e2|v>e2 + <e3|v>e3.
(4) The above obviously works for any dimensions. In fact, it works in infinitely-many dimensions, except in that case we have to be careful to make sure vectors have finite lengths.
Look up: "vector space" and "inner product" (very important), "Gram-Schmidt process" (helpful bonus)
(5) In the vector space of functions over some interval, <f|g> = Int[ f(x)g(x) dx ], the integral being over the domain. The only really important concern is that √<f|f> and √<g|g> must be finite. (Again, for complex-valued functions, conjugate f in the integral.)
(6) That over the the interval I = [-π,π], cos(nx) and sin(nx) have length-squared <c|c> = <s|s> = π for any integer n. Furtherfore, they're mutually all orthogonal.
(7a) Therefore, we can project any function over I as <cos nx|f>cos(nx)/π and <sin nx|f>sin(nx)/π. It is nontrivial (yet true) that these sines and cosines are all we need to represent a (sufficiently well-behaved) function fully.
(7b) f(x) = (1/π) Sum[ <cos nx|f>cos(nx) + <sin nx|f>sin(nx) ].
[Edit: typo of a missing > and a missing n and a slight re-wording to unambiguously refers to the projection in (2).]

Question for you: In the usual presentation of the Fourier series, the integrals <cos nx|f>/π are the a_n's. But a_0 has is usually written with an extra factor of 1/2 compared to the other a_n's. Where did that go?

One way of "getting to" the Fourier transform is to do a certain limit of the Fourier series with complex-valued functions; the summation above will then be a Riemann sum (i.e., an integral).

[geodesic42]

Per usual, as I remember, good post Vorpal. I figured I might 'high-school-ify' some of the things if that's alright. =)

VorpalNeko

Understanding the Fourier transform via the Fourier series first is a good approach. That's the f(x)cos(nx), and f(x)sin(nx) integrations referenced to above, since you would need to know enough calculus to do that anyway. It really helps in understanding how Fourier "picks out" the contribution of a particular frequency. It's literally a projection.

Explanation on projecting:
'Projection" is a good name for it. Think of 3 dimension space. The ground is 2 dimensional, and the space above it provides the third dimension. Picture a stick pocking out of the ground at an angle. That's like a vector. Now shine a light from above...the stick casts a shadow on the ground right below it! The shadow is like another vector: the projection of the stick onto the ground. So projecting something onto something else is like finding what part of the shadow of thing A touches thing B.

Quote:

A brief roadmap may be something like this:
...
(4) The above obviously works for any dimensions. In fact, it works in infinitely-many dimensions, except in that case we have to be careful to make sure vectors have finite lengths.
(5) In the vector space of functions over some interval, <f|g> = Int[ f(x)g(x) dx ], the integral being over the domain. The only really important concern is that √<f|f> and √<g|g> must be finite. (Again, for complex-valued functions, conjugate f in the integral.)

It might seem strange or confusing, but one can actually think of functions (rules that take one thing (say x) and associate it with another thing (say f(x))) in the same way as you do those 3 dimensional vectors...but there's a catch: it's an infinite dimensional space. Nonetheless, we should be able to choose those 'basis vectors' that Vorpal talked about and find the 'components' (the projections!) of a function vector along those basis functions.

Quote:

(7a) Therefore, we can project any function over I as <cos nx|f>cos(nx)/π and <sin nx|fsin(nx)/π. It is nontrivial (yet true) that these sines and cosines are all we need to represent a (sufficiently well-behaved) function fully.
(7b) f(x) = (1/π) Sum[ <cos nx|f>cos(nx) + <sin nx|f>sin(x) ].

A function that's periodic with period 2π. Obviously, not all functions are like that. In fact, most aren't periodic. Seems like we should be able to take the limit as the period goes to infinity though, right? Turns out, funny things happen to the vector space of functions then. The basis you'd want to choose becomes 'uncountable' (you can 'count' natural numbers - i.e. 1,2,3,4... - but can't count, for instance, the real numbers - i.e. 0, 0.000000.......(forever), 0.000000(forever)...(infinite number of things), 0.0001, (infinite number of other things), 1,...). So the sum over sin and cos become integrals. It turns out it's easiest to write these integrals with sin and cos in terms of e^(ix) type functions instead.

This is the long way of saying:

Quote:

One way of "getting to" the Fourier transform is to do a certain limit of the Fourier series with complex-valued functions; the summation above will then be a Riemann sum (i.e., an integral).

Happy trails!

[Analytica]

BUMP

[Captain Say]

www.khanacademy.org

This man is very, very articulate and covers the vast majority of calculus 1 and 2, along with a good introduction to linear algebra, which is the subject in which I was first introduced to fourier transforms though my professor is rather out of whack in terms of what to learn when. He's the only upper level professor at my college and teaches the same dozen or so students and just morphs our classes.

Differential equations is also very handy in there, haha.

Good luck!

[AnatidaeanAvenger]

This lecture series is good. The teacher is very clear and enthusiastic about the subject. This is aimed more towards electrical engineers than mathematicians, but it should help.
http://www.youtube.com/watch?v=gZNm7L96pfY
You should have some familiarity with complex numbers (e^ix = cos x + i sin x), periodic functions (periods/frequencies), sigma notation for sums, and integrating. Some linear algebra (vector spaces, projections, bases, inner products, etc.) makes the subject much more interesting, at least for me. Khanacademy is great for learning these.
They had a good set of lecture notes, but I can't find them anymore.