<p>Is there any research on harmonic analysis of sequences of Boolean functions, which represent the application of a Boolean function on a word in <span class="math-container">$(\{0,1\}^n)^*$</span>?</p>
<p>I'm looking for any reference on this, since my searches are not coming up with anything. Maybe I'm missing the right keywords.</p>
<p>To be more precise, consider a Boolean function <span class="math-container">$f:\{0,1\}^n\to \{0,1\}$</span>, and extend it to a function <span class="math-container">$f^*:(\{0,1\}^n)^*\to \{0,1\}^*$</span> by <span class="math-container">$f^*(w_0w_1\cdots w_m)=f(w_0)f(w_1)\cdots f(w_m)$</span>.</p>
<p>Now, for example, we can now measure the <em>influence</em> of a coordinate <span class="math-container">$i$</span> on <span class="math-container">$f^*$</span> by measuring the measure of words <span class="math-container">$x$</span> for which flipping the <span class="math-container">$i$</span>-th coordinate in <em>some</em> letter of <span class="math-container">$x$</span>, causes a change in the resulting word <span class="math-container">$f^*(x)$</span>.</p>
<p>Are such notions studied anywhere?</p>

CC BY-SA 4.0
2020-12-06T08:05:33.977Z
See related question:

https://mathoverflow.net/questions/377771/discrete-harmonic-analysis-with-infinite-unbounded-number-of-variables/378088#378088
kodlu
CC BY-SA 4.0
2020-12-06T08:25:44.923Z
Yeah, it's also one of mine, and indeed related. Unfortunately, the answer there doesn't resolve things.
Shaull
<reference-request><boolean-functions><fourier-analysis>
Harmonic analysis of sequences of Boolean functions (i.e. of words in $(\{0,1\}^n)^*$)
Shaull
# Harmonic analysis of sequences of Boolean functions (i.e. of words in $(\{0,1\}^n)^*$)

(Asked by: username_0 on 2020-09-15)

Is there any research on harmonic analysis of sequences of Boolean functions, which represent the application of a Boolean function on a word in $(\{0,1\}^n)^\*$?

I'm looking for any reference on this, since my searches are not coming up with anything. Maybe I'm missing the right keywords.

To be more precise, consider a Boolean function $f:\{0,1\}^n\to \{0,1\}$, and extend it to a function $f^\*:(\{0,1\}^n)^\*\to \{0,1\}^\*$ by $f^\*(w\_0w\_1\cdots w\_m)=f(w\_0)f(w\_1)\cdots f(w\_m)$.

Now, for example, we can now measure the *influence* of a coordinate $i$ on $f^\*$ by measuring the measure of words $x$ for which flipping the $i$-th coordinate in *some* letter of $x$, causes a change in the resulting word $f^\*(x)$.

Are such notions studied anywhere?

**Tags:** reference-request, boolean-functions, fourier-analysis
> **Comments:**
> **username_1**: See related question:  https://mathoverflow.net/questions/377771/discrete-harmonic-analysis-with-infinite-unbounded-number-of-variables/378088#378088  
> **username_0**: Yeah, it's also one of mine, and indeed related. Unfortunately, the answer there doesn't resolve things.  

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## 0 answers

