The Intuition Behind The Fourier Series and The Fourier Transform

The Fourier Transform and the Fourier Series are amongst the most profound mathematical discoveries of all time. They help us decompose functions into their basic ingredients. They reveal the fundamental blocks of any mathematical function and they enable us to use these blocks in order to better understand and manipulate them. But what exactly is the idea behind the Fourier Series and the Fourier Transform and what are these “fundamental ingredients”?

The Basic Idea

The intuition behind both the Fourier Series and the Fourier Transform is the same:

Any function can be written as the sum of sinusoidal functions.

An idea so simple yet extremely deep.

We were all taught what cosines and sines are in high school. They relate an angle of a right-angled triangle to ratios of two side lengths. Another way of understanding them is that cosine and sine are the x and y coordinates, respectively, of a point that moves around the unit circle. They are among the simplest periodic functions one can think of.