What Is the Fourier Transform?

Summary

As we listen to a piece of music, our ears perform a calculation. The high-pitched flutter of the flute, the middle tones of the violin, and the low hum of the double bass fill the air with pressure waves of many different frequencies. When the combined sound wave descends through the ear canal and into the spiral-shaped cochlea, hairs of different lengths resonate to the different pitches, separating the messy signal into buckets of elemental sounds. It took mathematicians until the 19th century to master this same calculation. In the early 1800s, the French mathematician Jean-Baptiste Joseph Fourier discovered a way to take any function and decompose it into a set of fundamental waves, or frequencies. Add these constituent frequencies back together, and you’ll get your original function. The technique, today called the Fourier transform, allowed the mathematician — previously an ardent proponent of the French revolution — to spur a mathematical revolution as well. Out of the Fourier transform grew an entire field of mathematics, called harmonic analysis, which studies the components of functions. Soon enough, mathematicians began to discover deep connections between harmonic analysis and other areas of math and physics, from number theory to differential equations to quantum mechanics. You can also find the Fourier transform at work in your computer, allowing you to compress files, enhance audio signals and more. “It’s hard to overestimate the influence of Fourier analysis in math,” said Leslie Greengard of New York University and the Flatiron Institute. “It touches almost every field of math and physics and chemistry and everything else.” Flames of Passion Fourier was born in 1768 amid the chaos of prerevolutionary France. Orphaned at 10 years old, he was educated at a convent in his hometown of Auxerre. He spent the next decade conflicted about whether to dedicate his life to religion or to math, eventually abandoning his religious training and becoming a teache...