Harmonic analysis on $\{0,1\}^{\mathbb{N}}$ can be different depending on how one **enumerates** orthonormal basis, so called *Walsh functions*.

One usually considers Haar measure $dm$ on $\{0,1\}^{\mathbb{N}}$ which is a probability measure in the following sense: if you have a function $f :\{0,1\}^{\mathbb{N}} \to \mathbb{R}$ which depends only on the first $n$ variables, say $f(x_{1}, ... ) =g(x_{1}, \ldots, x_{n})$ then
$$
\int_{\{0,1\}^{\mathbb{N}}} f(x)dm(x) = \frac{1}{2^{n}} \sum_{(x_{1}, \ldots, x_{n}) \in \{0,1\}^{n}}g(x_{1}, \ldots, x_{n}).
$$
Orthonormal system on $L^{2}(\{0,1\}^{\mathbb{N}}, dm)$ is called Walsh system, and perhaps it was first time introduced by Walsh in 1923: this is a certain family of functions on $\{0,1\}^{\mathbb{N}}$ taking values $+1$ or $-1$.

I have seen *4 different enumerations* of this family of functions used in the literature (perhaps there are more). The first 3 of them (original Walsh, Paley, and Walsh--Kaczmarz) is well described in the book I provided in the comments, see Section 1.4. The 4th one which I find kind of exceptional is not there.

**Walsh's original enumeration**: Walsh defined his system *recursively*. Later, it turned out that they can be written as follows:
$$
\varphi_{n}(x) = (-1)^{\sum_{k=0}^{\infty}(n_{k}+n_{k+1})x_{k+1}}, \quad n \in \mathbb{N},
$$
where $n=\sum_{k=0}^{\infty} n_{k} 2^{k}$ with $n_{k} \in \{0,1\}$, and $x=(x_{1}, \ldots ) \in \{0,1\}^{\mathbb{N}}$. In other words $(n_{0}, n_{1}, \ldots)$ is the binary representation of the positive integer $n$. The system $\{\varphi_{n}(x)\}_{n\geq 1}$ is the Walsh's original system.
Partial sums $S_{m}f = \sum_{k=0}^{m} c_{k} \varphi_{k}$, where $c_{k} = \int_{\{0,1\}^{\mathbb{N}}}f \varphi_{k} dm$, converge to $f$ in $L^{p}$, $p>1$. The latter is equivalent to the statement that
$$
\sup_{n \geq 1}\| S_{n} f\|_{p} \lesssim \|f\|_{p}, \quad 1<p<\infty.
$$
Moreover, $S_{n}f$ converges a.e. to $f$ for any $f \in L^{p}(\{0,1\}^{\mathbb{N}}, dm)$ provided that $p>1$. This is a consequence of Carleson's maximal inequality
$$
\|\sup_{n \geq 1} |S_{n} f|\|_{p} \lesssim \|f\|_{p}, \quad 1<p<\infty.
$$
Luzin asked in the trigonometric case on the unit circle $\mathbb{T}$. Carleson proved for $p=2$; Hunt extended to $1<p<\infty$. Billard proved for the original Walsh system for $p=2$; Sjölin extended for $1<p<\infty$.
Fejer means are bounded, i.e.,
$$
\sup_{n\geq 1}\left\| \frac{S_{1}+...+S_{n}}{n} f\right\|_{p} \lesssim \|f\|_{p}, \quad 1\leq p \leq \infty
$$
In the book I provided there is a more "martingale proofs" of these results.
Dyadic martingales naturally appear here. For example, $\{ S_{2^{k}}f\}_{k \geq 0}$ is a martingale. In fact, $S_{2^{k}}f = \mathbb{E} (f | \mathcal{F}_{k})$, where $\mathcal{F}_{k}$ is $\sigma$-algebra, which is generated by atoms, where atoms are $I_{k}(x) = \{ y \in \{0,1\}^{\mathbb{N}} : y_{1}=x_{1}, \ldots ,y_{k}=x_{k}\}$. In probabilistic language you condition on the first $k$ variables, and let the tail be the everything. Then, for example, the statement that
$$
\|\sup_{k \geq 0}|S_{2^{k}} f|\|_{p} \lesssim \|f\|_{p} \quad 1<p<\infty
$$
is exactly Doob's maximal inequality. The book F. Schipp, WR. Wade, P. Simon, "Walsh series: an introduction to dyadic harmonic analysis" provides more martingale techniques. The paper A guide to Calreson's theorem by Chiprian Demeter has more *time-frequency analysis* introduction.

**Paley's enumeration** (Walsh--Paley functions). In 1932, Paley proposed a different enumeration of Walsh functions
$$
w_{n}(x) = (-1)^{\sum_{k=0}^{\infty} n_{k}x_{k+1}}, \quad n \in \mathbb{N}.
$$
Paley's enumeration perhaps is more close to the classical trigonometric system $\sin(ns)$ and $\cos(ns)$. We can "identify" $[0,1)$ with $\{0,1\}^{\mathbb{N}}$ as follows:
$$
s \in [0,1), \quad s = \sum_{k=1}^{\infty} s_{k} 2^{-k}
$$
for some $s_{k} \in \{0,1\}$. This identification is not unique, for example, $\frac{1}{2^{2}}$ can be written as $(0,1,0,0,\ldots)$ and also as $(0,0,1,1,1\ldots,)$. Among these two one choose the first one, i.e., when number of 1's terminate starting from some place. Such *bad points* are not too many, they have zero measure so we do not bother with them. In particular $\omega_{n}(s)$ is defined for $s \in [0,1)$. Then
$$
w_{2^{k}}(s) = (-1)^{s_{k+1}} =\mathrm{sign}(\sin(2^{k+1} \pi s)) \approx \sin(2^{k+1} \pi s), \quad s \in (0,1].
$$
In general, $\omega_{n}(s)$ is close to $\sin(ns)$ or $\cos(ns)$ depending $n$ is even or odd.
Frequencies $\sin(2^{k+1} \pi s)$ are independent pretty much in the same way as $\omega_{2^{k}}(s) = (-1)^{s_{k+1}}$ are independent: Khinchin's inequality holds for both of them $\| \sum c_{k} \omega_{2^{k}} \|_{2} \asymp \| \sum c_{k} \omega_{2^{k}} \|_{1}$, and $\| \sum c_{k} \sin(2^{k+1} \pi s) \|_{1} \asymp \| \sum c_{k} \sin(2^{k+1} \pi s)\|_{2}$.
One has the notion of "dyadic derivative"
$$
df(x) = 2^{0} \frac{f(x_{1}, x_{2}, \ldots) - f(1-x_{1}, x_{2}, \ldots)}{2}+2^{1}\frac{f(x_{1}, x_{2}, \ldots) - f(x_{1},1-x_{2}, \ldots)}{2}+2^{2} \ldots
$$
provided that the sum converges. Perhaps one should call it "dyadic Laplacian" instead of the derivative. One can verify that Walsh--Paley functions are eigenfunctions: $d\omega_{n}(x) = n \omega_{n}(x)$. One can study polynomials of degree $m$, say finite sums $x \mapsto \sum_{k=0}^{d} c_{k}\omega_{k}(x)$, and ask questions similar to Markov--Brother's and Bernstein estimates for such polynomials. One can ask regularity questions: if $f$ is regular on $(0,1]$ how regular it is on $\{0,1\}^{\mathbb{N}}$ with respect to dyadic derivative and vice versa. What happens with partial sums if we have some regularity in terms of dyadic derivatives etc. The results about $L^{p}$ convergence of partial sums, Carleson--Hunt theorem also hold for this enumeration.
**Sneider's enumeration** (Walsh-Kaczmarz system) was introduced by Sneider:
$$
k_{0}(x) =1, \quad k_{n}(x) = (-1)^{x_{A}+\sum_{k=0}^{A-1} n_{A-k-1}x_{k+1}}\quad \text{where} \quad 2^{A}\leq n < 2^{A+1}
$$
I won't say much about this system except that the same $L^{p}$ convergence theorems hold in this case. Perhaps an interesting observation in all these 3 enumerations is that Dirichlet Kernels
$D_{n}(x) = \sum_{k=0}^{n-1} a_{n}(x)$ where $a_{n}$ is either Walsh, Walsh--Paley, Walsh--Kaczmarz, for $n=2^{k}$ are the same. In fact $D_{2^{k}}(x) =\prod_{j=1}^{k}(1+(-1)^{x_j})$.

**Frequencies:** when working with these 3 enumerations one usually thinks that $a_{n}(x)$ is "high-frequency" element if $n$ is large, and "low-frequency" element if $n$ is small. For example $\omega_{2^{k}}(x) = (-1)^{x_{k+1}}$ for large $k$ are high frequency elements similar to $\sin(2^{k+1}\pi s)$. All these 3 enumerations and *concepts of frequencies* tend to mimic the harmonic analysis on $L^{p}(\mathbb{T}, d\theta)$, where $d\theta$ is the uniform measure on the unit circle, i.e., trigonometric polynomials.

**Probabilistic enumeration** (*from independent towards dependent*). In this enumeration the concept of "high/low frequencies" is changed in a *very opposite* way. In fact, the elements $\{ \omega_{2^{k}}(x)\}_{k \geq 0}$ will be the lowest frequencies, say frequencies of degree $1$. Elements $\omega_{n}(x)$ where $n$ has two bits in its binary representation will be frequencies of order $2$. These are precisely Walsh functions $(-1)^{x_{i}+x_{j}}$, $i<j$. In a certain sense enumeration starts from "*independent frequencies to dependent ones*", and probabilistic ideas, measure concentration phenomena, dimension independent phenomena, and the concept of "independence" happen to be useful. The goal is to represent functions $f : \{0,1\}^{\mathbb{N}} \to \mathbb{R}$ as follows:
$$
f(x_{1},x_{2}, \ldots) = c_{0} + \underbrace{c_{1}(-1)^{x_{1}}+c_{2}(-1)^{x_{2}}+\ldots+c_{i}(-1)^{x_{i}}+\ldots}_{\mathrm{linear\; terms}}+\underbrace{c_{12}(-1)^{x_{1}+x_{2}}+c_{13}(-1)^{x_{1}+x_{3}}+\ldots+c_{ij}(-1)^{x_{i}+x_{j}}+\ldots}_{\mathrm{second\, order\, terms}}+\mathrm{higher\, order \, terms} \quad (1)
$$
Of course, if one takes $f \in L^{2}(\{0,1\}^{\mathbb{N}}, dm)$, writes into Walsh--Paley system $f(x) = \sum_{n\geq 0} a_{k} \omega_{k}(x)$, and wants to rearrange the terms to write in the form (1) then rearranged infinity sums may give different results unless $\sum |a_{n}| <\infty$ which is very unlikely for a "typical" $f \in L^{2}$. However, if one considers functions $f$ which depend only on the first $n$ variables the rearranged sums will be the same. This is not a big restriction on the class of functions because when one wants to prove a statement for functions on $\{0,1\}^{\mathbb{N}}$ one does not start with $f \in L^{2}$ which depends *on all its variables*, one starts from a "finite dimensional problem" when $f$ depends only on the first $n$ variables, and one proves the corresponding statements for such functions, requires the constants involved in the statement to be independent of $n$, and then passes to a limit, i.e., to an infinite dimensional problem.
Thus if we consider functions $f$ which depend only on the first $n$ variables then
$$
f(x_{1}, \ldots, x_{n}) = c_{0}+c_{1}(-1)^{x_{1}}+\ldots +c_{n}(-1)^{x_{n}}+\sum_{1\leq i<j \leq n} c_{ij}(-1)^{x_{i}+x_{j}} + \ldots \\= \sum_{S \subset \{1, \ldots, n\}} a_{S}\, (-1)^{\sum_{j \in S} x_{j}}
$$
The latter notation seems to be compact and convenient. Here $a_{S}$ are Fourier coefficients, and $(-1)^{\sum_{j \in S} x_{j}}$ are Walsh functions (enumerated in a strange way) indexed by the sets $S$. Since this enumeration works well provided that $f$ depends only on finite number of variables, the questions about $L^{p}$ convergence, Carleson--Hunt theorem etc do not make any sense. So the direction of harmonic analysis in which Luzin was interested ends here. Well, one can still consider *partial sums*
$$
S_{d} f (x) = \sum_{\substack{S \subset \{1, \ldots, n\}\\ |S|\leq d}} a_{S} \, (-1)^{\sum_{j \in S} x_{j}},
$$

where $|S|$ denotes the cardinality of the set $S$, and ask similar questions
$$
\| S_{d} f\|_{p} \lesssim \|f\|_{p}
$$
with uniform bound for all $d$, $1\leq d \leq n$, for all $n\geq 1$, and all $f$ depending on the first $n$ variables. Such estimate holds if and only if $p=2$.
Interesting connection arises with Gauss space. For example
$$
\frac{\ell!}{n^{\ell/2}} \sum_{\substack{S \subset\{1, \ldots, n\}\\ |S|=\ell}} (-1)^{\sum_{j \in S} x_{j}} \stackrel{d}{\to} H_{\ell}(\xi)
$$
where $H_{\ell}$ is degree $\ell$ probabilists' Hermite polynomial, $\xi \in \mathcal{N}(0,1)$ is the standard normal Gaussian random variable, and the convergence is in the sense of distributions. By Taking tensor products of the above example one recovers Hermite polynomials on $\mathbb{R}^{N}$ with arbitrary $N$. So, because of this limit, it is reasonable to think about $(-1)^{\sum_{j \in S} x_{j}}$ as low frequency element if $|S|$ is small, and high frequency element if $|S|$ is large (which is very opposite compared to the original Walsh enumeration). In fact, the analysis with such enumeration (1) for functions $f$ depending on $n$ variables is similar to the one on $L^{p}(\mathbb{T}^{n}, d\theta^{n})$. *Dscrete Laplacian* is defined in a different way
$$
\Delta f(x) = \sum_{j=1}^{n} D_{j}f(x),
$$
where $D_{j}f(x_{1}, \ldots, x_{n}) = \frac{f(x_{1}, \ldots,x_{j}, \ldots, x_{n})-f(x_{1}, \ldots, 1-x_{j}, \ldots, x_{n})}{2}$. One also considers discrete gradient
$$
Df(x) = (D_{1} f(x), \ldots, D_{n}f(x)).
$$
One has *integration by parts formula*
$$
\int_{\{0,1\}^{\mathbb{N}}} g \Delta f dm=\int_{\{0,1\}^{\mathbb{N}}} Df \cdot Dg\, dm
$$
for all functions $f,g$ depending on the first $n$ variables. Heat semigroup

$$
e^{-t\Delta} f = \sum_{S \subset \{1, \ldots, n\}}a_{S} e^{-t|S|} (-1)^{\sum_{j\in S}x_{j}}
$$
satisfies heat equation $\frac{\partial }{\partial t} e^{-t\Delta}f = -\Delta e^{-t\Delta}f$. Discrete gradient
$$
|\nabla f|(x) = \sqrt{\sum_{j=1}^{n} |D_{j}f(x)|^{2}}
$$
when applied to $f(x_{1}, \ldots, x_{n}) = g\left(\frac{(-1)^{x_{1}}+\ldots+(-1)^{x_{n}}}{\sqrt{n}}\right)$ for some smooth bounded $g :\mathbb{R} \to\mathbb{R}$, converges to the classical derivative

$$
\int_{\mathbb\{0,1\}^{\mathbb{N}}} M(|\nabla f|(x)) dm(x) \stackrel{n \to \infty}{\to} \int_{\mathbb{R}}M(|g'(s)|) \frac{e^{-s^{2}/2}}{\sqrt{2\pi}}ds
$$
for any smooth functions $M$. The case $M(t)=|t|^{p}$ is a typical application. Denoting $(-1)^{x_{j}} = \varepsilon_{j}$, then *$\ell$ degree polynomials* with respect to this enumeration can be written as
$$
g(\varepsilon_{1}, \ldots, \varepsilon_{n}) =\sum_{\substack{S \subset \{1, \ldots, n\} \\ |S|\leq \ell}} a_{S} \prod_{j \in S} \varepsilon_{j}
$$
and the latter expression can be extended in a natural way for all $\varepsilon_{1}, \ldots, \varepsilon_{n} \in \mathbb{R}$ as a multilinear classical polynomial in $n$ variable of degree $d$. Due to the identity $\| g\|_{L^{\infty}([-1,1]^{n})} = \| g\|_{L^{\infty}(\{-1,1\}^{n})}$ tools from approximation theory enter in an unexpected way applied to actual degree $\ell$ multilinear polynomials. Discrete derivatives coincide with the actual derivatives for such multilinear polynomials. Boolean functions $f :\{0,1\}^{\mathbb{N}} \to \{0,1\}$ have an extra property $f^{2}=f$ and this gives more cancelations when proving statements for all real valued functions.

To summarize: Harmonic analysis on $\{0,1\}^{\mathbb{N}}$ depends on how one encodes Walsh functions, i.e., *frequencies* in the Hilbert space $L^{2}(\{0,1\}^{\mathbb{N}}, dm)$, who are the low and who are the high frequency functions.

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