Vectors and Fourier Series: Part 2

In Part 1, we developed three tools: a Basis in which to represent our vectors, Projection Operators to find the components of our vector, and a Vector Rebuilder which allows us to recreate our vector in the new basis. This is the second post is a series of three:

  • Part 1: Developing tools from vectors
  • Part 2: Using these tools for Fourier series
  • Part 3: A few examples using these tools

We now want to develop these tools and apply the intuition to Fourier Series. The goal will be to represent a function (vectors) as the sum of sines and cosines (our basis). To do this we will need to define a basis, create projection operators, and create a functions rebuilder.

We will restrict ourselves to functions on the interval: \([-L,L]\). A more general technique is the Fourier Transform, which can be applied to functions on more general intervals. Many of the ideas we develop for Fourier Series can be applied to Fourier Transforms.

Note: I originally wrote this post with the interval \([0,L]\). It's more standard (and a bit easier) to use \([-L,L]\), so I've since changed things to this convention. Video has not been updated, sorry :/

Choosing Basis Functions

Our first task will be to choose a set of basis function. We have some freedom to choose a basis as long as each basis function is normalized and is orthogonal to every other basis function (an orthonormal basis!). To check this, we need to define something equivalent to the dot product for vectors. A dot product tells us how much two vectors overlap. A similar operation for functions is integration.

Let's look at the integral of two functions multiplied together over the interval: \([-L, L]\). This will be our guess for the definition of a dot product for functions, but it is just a guess.

\(\int_{-L}^Ldx~f(x)g(x)\)

If we imagine discretizing the integral, the integral becomes a sum of values from \(f(x)\) multiplied by values of \(g(x)\), which smells a lot like a dot product. In the companion video, I'll look more at this intuition.

Now, we get to make another guess. We could choose many different basis functions in principle. Our motivation will be our knowledge that we already think about many things in terms of a frequency basis, e.g. sound, light, planetary motion. Based on this motivation, we'll let our basis functions be:

\(s_n(x) = A_n \sin(\tfrac{n\pi x}{L})\)

and

\(c_n(x) = B_n \cos(\tfrac{n\pi x}{L})\).

We need to normalize these basis functions and check that they are orthogonal. Both of these can be done through some quick integrals using Wolfram Alpha. We get

\(s_n(x) = \tfrac{1}{\sqrt{L}} \sin(\tfrac{n\pi x}{L})\)

and

\(c_n(x) = \tfrac{1}{\sqrt{L}} \cos(\tfrac{n\pi x}{L})\).

This is a different convention from what is commonly used in Fourier Series (see the Wolfram MathWorld page for more details), but it will be equivalent. You might call what I'm doing the "normalized basis" convention and the typical one is more of a physics convention (put the \(\pi\)s in the Fourier space).

Projection Operators

Great! Now we need to find Projection Operators to help us write functions in terms of our basis. Taking a cue from the projection operators for normal vectors, we should take the "dot product" of our function with the basis vectors.

\(\text{Proj}_{s_n}(f(x)) = \int_{-L}^L dx~\tfrac{1}{\sqrt{L}} \sin(\tfrac{n\pi x}{L})f(x)=a_n\)

and

\(\text{Proj}_{c_n}(f(x)) = \int_{-L}^L dx~\tfrac{1}{\sqrt{L}} \cos(\tfrac{n\pi x}{L})f(x) = b_n\).

Vector Rebuilder

Now we can rebuild our function from the coefficients from the dot product multiplied by our basis functions, just like regular vectors.

\(f(x)=\sum\limits_{n=0}^\infty\text{Proj}_{s_n}(f(x))+\text{Proj}_{c_n}(f(x))\\ = \sum\limits_{n=0}^\infty \int_{-L}^L dx~\tfrac{1}{\sqrt{L}} \sin(\tfrac{n\pi x}{L})f(x)+\int_{-L}^L dx~\tfrac{1}{\sqrt{L}} \cos(\tfrac{n\pi x}{L})f(x)\\ = \sum\limits_{n=0}^\infty a_n \tfrac{1}{\sqrt{L}} \sin(\tfrac{n\pi x}{L})+b_n \tfrac{1}{\sqrt{L}} \cos(\tfrac{n\pi x}{L})\)

To recap, we've guessed a seemingly useful way of defining basis vectors, dot products, and projection operators for functions. Using these tools, we can write down a formal way of breaking down a function into a sum of sines and cosines. This is what people call writing out a Fourier Series. In the think post of the series, I'll go through a couple of problems so that you can get a flavor for how this pans out.

Youtube companion to this post:

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