Over the years I have gotten used to seeing many theorems in theoretical computer science being proved using discrete Fourier analysis. The Walsh-Fourier (Hadamard) transform of Boolean functions proved to be extremely useful in virtually every subfield of theoretical computer science, including PCPs, property testing, pseudorandomness, and communication complexity. As it turns out, many seemingly hard problems can be solved by writing the Walsh-Fourier expansion and using basic theorems of harmonic analysis.

While I have gotten quite comfortable using Fourier analysis when trying to tackle a problem, and even though I developed a pretty good hunch for which cases Fourier analysis should yield nice results – I have to admit that it took me a long time to begin to unravel the Fourier magic; that is, to understand what is so special about the Walsh-Fourier basis that makes it so powerful.

For the rest of this post, I assume that the readers have some proficiency in Fourier analysis. However, in order to make this post (hopefully) a bit more accessible to readers who are new to harmonic analysis, I would like to dedicate the rest of this paragraph to stating out some basic definitions and facts regrading the Fourier transform. Say we have a real function on the hypercube $latex {f:\mathbb{Z}_2^n \rightarrow \mathbb{R}}&fg=000000$. The *Walsh-Fourier expansion* of $latex {f}&fg=000000$ is defined as follows:

$latex \displaystyle f(x) = \sum_{S \subseteq [n]}\hat{f}(S)\chi_S(x), \ \ \ \ \ (1)&fg=000000$

where the characters $latex {\chi_S: \mathbb{Z}_2^n \rightarrow \mathbb{R}}&fg=000000$ are defined by

$latex \displaystyle \chi_S(x_1, \ldots, x_n) = (-1)^{\sum_{i \in S} x_i} \ \ \ \ \ (2)&fg=000000$

(note that the summation in Eq. (2) is over $latex {\mathbb{Z}_2^n}&fg=000000$, and that the coefficient of each character $latex {\chi_S}&fg=000000$ is $latex {\hat{f}(S) = \langle f, \chi_S \rangle}&fg=000000$). The way I like to think of the Fourier transform is simply as a change of basis for $latex {f}&fg=000000$, wherein (hopefully) the representation of $latex {f}&fg=000000$ is simpler, and hence easier to analyse. It is also interesting to note that if $latex {f}&fg=000000$ is defined on $latex {\{-1,1\}^n}&fg=000000$ instead on $latex {\mathbb{Z}_2^n}&fg=000000$, then each character $latex {\chi_S: \{-1,1\}^n \rightarrow \mathbb{R}}&fg=000000$ is the monomial $latex {\chi_S(x) = \prod_{i \in S} x_i}&fg=000000$, thus the Fourier expansion is in fact the representation of $latex {f}&fg=000000$ as a multilinear polynomial over $latex {\mathbb{R}}&fg=000000$.

So sure, we have a relatively good understanding of how polynomials behave, and as the foregoing discussion shows, the Fourier transform is a natural way of looking at a function as a multilinear polynomial. But why specifically this base? What is so unique about the base of parities? Are there other bases which are just as effective? Here is my point of view, which I learned from Or Meir: Let $latex {\mathcal{F}}&fg=000000$ be the linear space of functions $latex {f: \mathbb{Z}_2^n\rightarrow\mathbb{R}}&fg=000000$. For every $latex {w\in\mathbb{Z}_2^n}&fg=000000$, consider the linear operator $latex {\sigma_w}&fg=000000$ that maps a function $latex {f \in \mathcal{F}}&fg=000000$ to the function $latex {f’ \in \mathcal{F}}&fg=000000$ such that $latex {f'(x) = f(x+w)}&fg=000000$. Such operators are called *shift operators*. It turns out that in many natural problems that appear in theoretical computer science (but also in functional analysis, Hardy spaces, the theory of abelian varieties, and the theory of symbolic dynamics) there is a fundamental, underlying need to analyze the effects that such operators have on Boolean functions. Now, a cool property of the Fourier basis (namely, the shift theorem) is the fact that it simultaneously diagonalizes all of the shift operators. Details follow.

Since we are dealing with a finite discrete domain (the Boolean hypercube), we can view the functions in $latex {\mathcal{F}}&fg=000000$ as vectors in a subspace (i.e, $latex {f\in \mathcal{F}}&fg=000000$ can be viewed as a vector $latex {v_f \in \mathbb{R}^{2^n}}&fg=000000$, where the entries of $latex {v_f}&fg=000000$ are the values of the truth table of $latex {f}&fg=000000$). Then, the shift operators are linear operations on vectors, hence they can be viewed as matrices. As we mentioned before, we wish to simplify the representation of the functions and operators that we study, in order to make them easier to analyse. The best we can hope for is to diagonalize the matrix of the operator we inspect, and this is exactly what the Fourier basis does: In the Fourier basis, all of the shift operators are diagonal matrices.

More generally, the Fourier basis diagonalizes the convolution operator, which also underlies the structure of many natural problems in the analysis of Boolean functions. To see the power of the aforementioned diagonalization, let’s look at an important corollary of it: the *convolution theorem*. Given functions $latex {f,g \in \mathcal{F}}&fg=000000$, their *convolution* $latex {f * g}&fg=000000$ is also a function in $latex {\mathcal{F}}&fg=000000$, defined by

$latex \displaystyle [f*g](x) = \sum_{y \in \mathbb{Z}_2^n}f(y)g(x-y). \ \ \ \ \ (3)&fg=000000$

The convolution theorem for $latex {\mathbb{Z}_2^n}&fg=000000$ states that the Fourier transform of the pointwise multiplication of two functions is equal to the convolution of their Fourier transforms; that is,

$latex \displaystyle \widehat{f \cdot g} = \hat{f} * \hat{g}. \ \ \ \ \ (4)&fg=000000$

The convolution theorem, as well as the other aforementioned properties of the Fourier transform, makes it very robust for the analysis of Boolean functions. Let me provide an example from my own work (joint with Omer Tamuz). We want to test whether a function $latex {f \in \mathcal{F}}&fg=000000$ is Boolean or not (i.e, whether its image is contained in a set of cardinality 2, say $latex {\{-1,1\}}&fg=000000$). A simple, though crucial, observation is that a function $latex {f \in \mathcal{F}}&fg=000000$ is Boolean if and only if the convolution of $latex {f}&fg=000000$ with itself (which sometimes referred to as the *autocorrelation* of $latex {f}&fg=000000$) is equal to the delta function (i.e., the function that gets 1 at 0, and gets 0 elsewhere). To see this, note that $latex {f}&fg=000000$ is Boolean if and only if $latex {f^2 = 1}&fg=000000$, apply the convolution theorem, and use the fact that the Fourier transform of the constant 1 function is the delta function. Hence, using the Fourier expansion we are able to give a characterization of Boolean functions that is efficiently testable. This is a general theme: whenever a function is “correlated with its shifted-self”, it begs for a Fourier based approach.

Last, I would like to give a brief taste of “heavier” tools in the same spirit: We can generalize the discussion on efficient representations even further. Fourier analysis is a special case of the representation theory of finite groups. This theory deals with the more general space of functions $latex {f:G\rightarrow \mathbb{C}}&fg=000000$ where $latex {G}&fg=000000$ is a finite group. It allows us to find a basis that makes the analysis of shift operators easier, even though for general groups the shift operators aren’t always diagonalizable. Another possible generalization can be done by analyzing tuples of shifts (e.g., $latex {f(x),f(x+w_1),f(x+w_2),f(x+w_1+w_2)}&fg=000000$). In many cases, such analysis cannot be done by standard Fourier analysis, and higher-order Fourier analysis is needed (for more details, see Terry Tao’s excellent survey on this topic).

**Open Problems:** Returning to the original question at the beginning of the post: are there other bases of functions that will work as well as the Fourier basis? Can we perhaps mimic the success of wavelets in classical harmonic analysis, and use such bases in theoretical computer science?