In Conversation with Dr. Henry Cohn: Three Perspectives on the Fourier Series
The Fourier Series seem to pop up everywhere in math and science
Whether you are an 18.03 student or a math enthusiast, chances are you have heard about the Fourier series. Though a vital tool in math, physics, and beyond, it remains a topic that eludes intuition, even for those who have studied it in class. The Tech sat down with Professor Henry Cohn, who taught Differential Equations (18.03) in the fall of 2024, to offer three perspectives on the Fourier Series. Regardless of who you are or what you study, this is a glimpse into the beauty and power of this tool.
For the General Audience
Starting with the most fundamental intuition—the level of math without math—the Fourier Series may be best understood through music. On a piano, for instance, it would be dull to play only single notes one at a time. Instead, chords can produce more harmonious sounds. Professor Cohn recommended thinking about the Fourier series in the context of sound waves, where “you can have very complex sounds and you can break them apart into combinations of much simpler frequencies.” This process is analogous to the piano, which, in a sense, discretizes musical pieces into pure tones; the set of only 88 keys is a far simpler set of sounds compared to a piece like “La Campanella.” The Fourier Series is the sum of many different simple sine waves, which are periodic, into a more complicated periodic function, which involves the repetition of the same motif.
However, is it possible to represent non-repeating sounds like human speech? As it turns out, the Fourier Series continues to be useful: as Cohn recommended, “You can take a snippet of the conversation," and play on a loop to make it periodic, which is "really equivalent to doing the Fourier series, where you repeat the snippet over and over." Alternatively, the Fourier series can be generalized into one infinitely long, aperiodic motif through a deeper topic called Fourier Transforms. They are still modeled using sines and cosines, but now, they are stretched to infinity in both directions so that there are no repetitions. Science Youtuber and former NASA engineer Mark Rober delivered a fascinating example by synthesizing speech with different combinations of notes on a piano.
For the Pragmatist
You may wonder why it is useful at all to decompose a function into simpler parts when we can just have the function itself. On a mathematical level, Fourier series are often easier to manipulate than complicated functions: they are either written as sums of complex exponentials or trigonometric functions, which are straightforward to integrate or derive. On the application level, the methods of the Fourier series are used in the real world on tasks such as breaking down complex waves in audio processing and analyzing molecules through X-ray diffraction.
Another surprising application is solving the heat equation, which was in fact the initial problem that gave rise to the Fourier Series that French mathematician Joseph Fourier faced in 1822. The heat equation models heat distribution across a metal rod with both ends submerged in ice. “It doesn’t look like it has anything to do with representing periodic functions out of pure sine waves. And so the fact that this turns out to be exactly the mathematics underlying this is kind of a miracle,” Cohn said.
To get a sense of why this works, consider θ(x, t), a function that models the temperature θ of a piece of the metal rod at position, x, and time, t. The change in heat flux along the position on the rod is proportional to the rate of cooling with respect to time by a constant factor. Or, in symbols, t∝2x2.
Fourier’s solution, which involves doing separation of variables by setting
θ(x, t) = v(x)w(t), leads to this Fourier series: n=1∞bne(-n2t)sin(nx) for boundary conditions θ(0, t) = θ(𝜋, t) = 0.
In case you’re wondering where the n2 term in the exponential comes from, one explanation is that this equation connects to the number theoretic problem of “generating functions that tell you things like how many ways there are to write an integer as a sum of a certain number of squares.” Both the heat equation and the formula for generating functions that are sums of squares come from the common theta (θ) function, which is why theta notation is being used here.
The math behind the heat equation and breaking up whole numbers into sums of squares converges at a bigger mathematical topic called theta functions, which heavily involve the Fourier series: these infinite sums of sines and cosines are truly a stepping stone to higher realms of STEM.
For the Math Theorist
The wonders of the Fourier Series are not limited to its applications. As Dr. Cohn stated, “Everything is connected behind the scenes.” In fact, the Fourier series studied for their real-world applications are often also some of the most interesting to pure mathematicians.
From a mathematical perspective, we may naturally wonder what the space of possible manipulations may be after decomposing these functions, and why some functions are designated as “simple” functions. This is a specialized field of study called representation theory. Returning to the example of audio, one of the easiest ways to process it is to shift its entirety in time. Cohn added, “Here you can think of this as having a group acting on it, where the group is just one-dimensional. It's just the addition of real numbers or one-dimensional vectors, which is a sort of fundamental signal processing tool.”
Further, the behavior of such groups can be studied through an algebraic lens by finding the characteristic functions of the transformation that remain, in some sense, invariant, which are called eigenfunctions. More concretely, we ask when a function has the property that shifting it over in time is equivalent to multiplying it by a scalar. “The exponential functions are exactly the eigenfunctions for this,” Cohn explained, “but real exponentials blow up in one direction or another, so they're not really useful for building well-behaved signals. But the complex exponentials never get bigger or smaller; they always stay the same absolute value.” The one-dimensional Fourier transform uses these complex eigenfunctions to build almost any signal.
Of course, there are functions with more variables, and transformation groups more complicated than time shifts. Cohn offered a function with three spatial variables, which similarly works with complex exponential eigenfuncti instead of using the three-dimensional Fourier transform. There are also “weirder” transformations such as modular forms, discrete transformations of data sets symmetric under permutation groups, and more. The big picture of representation theory, Cohn summarized, is to understand the type of problem by figuring out what the underlying group of symmetries is, then analyzing it and finding its representations to characterize the original problem.
“In exciting cases, you'll end up near the research frontier,” he said.