Fourier Transform

Fourier Series

We can write any periodic function over a period as

  • If , it simplifies the expression

with coefficients are given by solving the following integrals:

Now, given the above sinusoidal expression of a function (with period )

we can express it more concisely using Euler’s Formula as where the coefficients are complex numbers.

But can you express c_k as a sum of a_k and b_k?

Yup.

Approximation

An approximation of a function could be achieved by truncating the series to a finite number of sinusoids:

For points (assume is even), we will use degrees of freedom (i.e., coefficients) to exactly interpolate the data.

Plugging in each of our data points into the expression will give us equations, involving unknowns coefficients, . This leads us to the Discrete Fourier Transform!

From MATH213

Definition 1: L^2 functions

A complex valued function is in the class if
exists and is finite.

is in the class if

exists and is finite.

We want to write for

Our goal is to find .

Definition 3: Fourier Series - Complex Form

If then the Fourier series in complex form of is where the are found by projecting into the basis of complex exponentials.

Theorem 3: Fourier Coefficients for Series in Complex Form

If then the Fourier coefficients of are

If is real valued than .

Theorem

For a real valued, periodic function :

  • If is an even then the Fourier series can be simplified to a sum of waves.
  • If is an odd then the Fourier series can be simplified to a sum of waves.

Theorem 2: Fourier Sine and Cosine Coefficients

If is a real valued function that is in then

If is even then the Fourier cosine series for is where

If is odd then the Fourier sine series for is

where

  • It’s a little hard to memorize this. Instead, memorize theorem 3 and derive it

Convergence of Fourier Series

See PWC1.

Theorem 5: Convergence of Fourier series

Let be the Periodic Extension of a function .

  • The Fourier series of converges in the norm (also in the mean and almost everywhere) to (and also ) on any finite subinterval of .
  • If is piecewise then the Fourier series of converges pointwise to for all
  • If is piecewise and continuous then the Fourier series of converges uniformly on any finite interval of .