Understanding Fourier Series: Conceptual Explanation and Calculation Tips

[jwsiii]

I'm having a hard time grasping exactly what a Fourier series is. I know the book definition and that it represents any periodic function as an infinite series. I also know that the a0 term is the average value of the function over one period. I can calculate the terms and everything but I don't really understand what I'm doing; I just know how to go through the calculations. Can someone give me a good, conceptual explanation of what a Fourier series is and how it works? Thanks.

 

[James R]

A Fourier series expresses any function as a sum of sine and cosine waves of different frequencies. For example, if you're talking about sound, you can take a wave of any shape you like and extract the frequency components of that wave by expressing it as a Fourier series.

 

[Crosson]

This my sound strange to you, but I think it may also be the best way to anser your question.

Presumambly you have worked with euclidean vectors: Arrows (points) in 3-space and 2-space. You may be familiar with an operation referred to as a scalar product, or dot product. If we take the product A dot B, we get the component of A in the direction of B.

It is possible to generalize generalize the idea of vectors to include functions, and to generalize the dot product as an integral. Essentially, what you are doing by taking the nth Fourier integral is finding the component of f[x] in the Sin[n x] direction.

 

[jwsiii]

Earlier in the math course I'm taking now, we did a block on vector calculus and learned the flux, which is the integral over a surface of the dot product of the vectors and the normal vector in a vector field. Is that similar to what's going on in a Fourier series in any way?

Also, I kind of understand the representation of a function as a vector but not quite. Is it like having a vector that points either straight up or down in the y-direction for each point on the x-axis? Thanks.

 

[mathwonk]

A vector in the plane is an arrow that points from the origin to some other point (x,y).

it can be written as a "sum" of two simpler vectors, simpler because they are either horizontal or vertical. to find the terms in this sum, we ask of our vector (x,y) how much it points in the hporizontal direction, [answer (x,0)], and how much of it points in the vertical direction, [answer (0,y)].


one way to tell that a vector such as (0,6) does not point at all in the hporizontal direction is to note that it is perpendicular to the standard horizontal vector (1,0).

since dot products measure perpendicularity, they help decompose a vector into horizontal and vertical parts.

To say (0,6) is perpendicular to (1,0), means that as functions with only two values, they take on their values on different points in the domain. i.e. (0,6) is "supported" only on the second entry while (1,0) is supported on the first entry. so when we multiply their values: 0.1 + 6.0, we just get zero.

we also get zero when their values on these points cancel each other out, as in (2,1) and (-1,2), so the dot product is 2(-1) +(1(2).


to tell whether a function f is supported on the same or opposite points of its domain from another standard function like sin(nx) we also multiply them f(x)sin(nx), and then see how big this product is by integrating it.

for instance if f is large where sinj(nx) is small the integral should be small, and also if the values of f cancel out those of sin(nx).

Since vectors are just functions which have only two domain points, this idea of perpendicularity of functions is a direct generalization of that of vectors, and allows us to decide how much f points in the direction of sin(nx), and hence to write f as a sum of various standard functions sin(nx), cos(mx).

note we have to choose different standard functions here as functions like (1,0,...,0) with only one non zero entry are not detectable by integrating.


Actually sin and cos are not the best choices, better are e^nx and e^inx , since these standard functions are eigenfunctions for the operator D of differentiation.

of course sinx = (1/2i)[e^(x) - e^(ix)] and cos(x) = (1/2)(e^x + e^(ix)), so the distinction is not fatal.

 

[fourier jr]

there's a theorem that says: let $${x_1, x_2, ...}$$ be an orthonormal set in an inner product space V & let $$x \in V$$.
spose there exist scalars $$c_1, c_2, ... ,c_n$$ such that $$x = \sum_{k=1}^{n}c_k x_k$$ then $$c_k = <x,x_k>$$ for k=1,...,n

that sum thing is the Fourier series of the vector x. it's just a different way of writing it. that <x, x_k> bit is the inner product, which is just the dot product of 2 vectors. orthonormal set means you've got a bunch of unit vectors that are all perpendicular to each other.

the definition of Fourier series goes something like this:

let $$X = {x_1, x_2, ...}$$ be a countable orthonormal set in an inner product space V & let $$x \in V$$. the infinite series $$\sum_{n=1}^{\infty}<x, x_k>x_k$$ is called the Fourier series of x, & the coefficient <x, x_k> is called the Fourier coefficient of x.

that probably doesn't make much sense to a 2nd-yr or 3rd-yr student, but a countable orthonormal set could just be the standard basis for V=R^2: { (0,1), (1,0) } & the inner product would be the usual dot product. you can have different inner products & different "vectors" though; one inner product is the standard formula for Fourier coefficients: $$\frac{1}{L} \int_{L}^{L+2L}f(x)sin(\frac{n\pi x}{L})dx$$ (similarily for cosine series), & the "vectors" are sines & cosines, etc.

 

[mathwonk]

that "theorem" is essentially distributivity of multiplication.