Mathematics is often referred to as the abstract science of numbers, quantities, and space. The power of mathematics lies in its wide range of applicability to various different fields. At the core of our modern lives lies the concepts mathematicians have spent thousands of years perfecting. In what follows I would like to discuss some of the beauty and elegance that lies within the Fourier Series and its broader applications. However, before we get to the details about Fourier Series it would be beneficial to define elegance and beauty in our context.

**Elegance**: Merriam-Webster defines elegance as the tasteful richness of design.

**Beauty: **Merriam-Webster defines beauty as the quality or aggregate of qualities in a “thing” that gives pleasure to the senses or pleasurably exalts the mind or spirit.

Mathematical concepts such as the Fourier Series display qualities that can be categorized as beautiful and or elegance. The main idea of the Fourier Series is to break down functions and patterns as combinations of simple oscillations. Fourier Series is named after its pioneer Joseph Fourier. The series can be defined as the expansion of a period function using the infinite sum of sines and cosines. The initial challenge that Fourier was aiming to solve with his findings was the propagation of heat. However, later it was discovered that the techniques developed by Joseph can be used to solve problems that involve linear differential equations with constant coefficients. As it turns out linear differential equations with constant coefficients show up in fields such as electrical engineering, vibration analysis, acoustics, optics, signal and image processing, quantum mechanics, economics, and many more.

Being able to break down complex oscillations and treat them as a summation of sines and cosines has had a tremendous effect on advancing our understanding of the universe. One of the most prominent areas where ideas introduced by Fourier shine is in mobile communication systems. Fourier’s tasteful richness of design laid the groundwork for telecommunication engineers to be able to transfer massive amounts of data over networks.

For example, sending a square pulse wave over a communication channel would require infinite bandwidth which is not possible to accomplish with the tools and knowledge we currently have at hand. Fourier transform comes in handy to solve this problem. In figure 1 the red line is the Fourier approximation of square pulse wave which can be drawn using the function below.

The figure and formula above were taken from here.

If you’d like a more visual explanation of the Fourier Series I’d recommend But what is a Fourier series? by 3Blue1Brown.