# Questions tagged [fourier-analysis]

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### Why does Fourier analysis of Boolean functions “work”?

Over the years I have gotten used to seeing many TCS theorems proved using discrete Fourier analysis. The Walsh-Fourier (Hadamard) transform is useful in virtually every subfield of TCS, including ...
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### Applications of representation theory of the symmetric group

Inspired by this question and in particular the final paragraph of Or's answer, I have the following question: Do you know of any applications of the representation theory of the symmetric group in ...
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### Fourier coefficients Boolean Functions described by Bounded Depth Circuits with AND OR and XOR gates

Let $f$ be a Boolean function and let's think about f as a function from $\{-1,1\}^n$ to $\{ -1,1 \}$. In this language the Fourier expansion of f is simply the expansion of f in terms of square free ...
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### Is there a simple argument for this Hemi-Icosahedron Boolean function?

This is problem 1(e) from Homework 1 of the course about Analysis of Boolean functions at CMU in 2012 as well as problem 1.1(n) on p.34 of Ryan O'Donnell's Analysis of Boolean Functions. Compute ...
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### Fourier Analysis on checking whether there exists a vector in hypercube orthogonal to a set of vectors?

I know virtually noting about Fourier Analysis and I'd like to know whether it's worth to learn this topic for my problem. My problem is: Given vectors $h_1,\cdots,h_k\in\{+1,-1\}^n$ where $k < n$,...
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### Relationship between sparsity and rank of a boolean function

I have the following question when I was going through the proof of the following theorem. Theorem. For XOR function $f \circ XOR$, $rank(M_{f \circ XOR}) = ||\hat f ||_0$ where $M_{f \circ XOR}$ is ...
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### About counting the “total size” of non-zero Fourier coefficients of a Boolean function

Given a $f: \{-1,1\}^n \rightarrow \mathbb{R}$, I want to compute this quantity, $\sum_{ \hat{f}(S) \neq 0, S \subseteq 2^{[n]}} \vert S \vert$ i.e the sum of the sizes of the subsets of $[n]$ ...
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### How to prove deg(f) = n iff the parity imbalance of f is non-zero?

Not sure if the notation I'm using here is standard or not. I'm going over class notes and I'm stumped over an exercise given: Show that $deg(f) = n \iff PI(f) \neq 0$. Here $f$ is a boolean function ...
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### Basic property of boolean functions with restrictions

For $f:\{\pm1\}^n\to\mathbb{R}$, $I\subset\{1,\dots,n\}$ and $x\in\{\pm1\}^{\{1,\dots,n\}\setminus I}$ we define $f_I[x]:\{\pm1\}^I\to\mathbb{R}$ by $f_I[x](y)=f(x,y)$. (We denote by ($x,y$) the ...
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### Given a subset of of the hypercube and an affine transform of it, find the affine map

This is a follow up to this resolved question. Suppose we are given a set of bitvectors $A\subseteq\mathbb{F}_2^d$ and an invertible affine transformed copy of it $$B=\{Mx + s\mid x\in A\}$$ for some ...
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### Lower bound on the support size of an $\epsilon$-biased distribution

Let $D$ be an $\epsilon$-biased distribution we want to show that $$\text{Supp}(D)\geq \Omega\bigg(\frac{n}{\epsilon^2\log(\frac{1}{\epsilon})}\bigg)$$ I know that there are some proofs for this but I ...
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### Nearly Optimal Sparse Walsh-Fourier Tranform

In the recent paper: Nearly Optimal Sparse Fourier Transform[Haitham Hassanieh, Piotr Indyk, Dina Katabi, Eric Price], the authors show an $O(k \log n)$-time algorithm for the problem of computing the ...
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### Level $k$ bounds in Analysis of Boolean functions

In Ryan O'Donnell's book Analysis of Boolean functions, following Corollary 9.25 the following appears: If $f\colon \{-1,1\}^n \to \{0,1\}$, and we have $\mathbb{E}[f] = \alpha$, then for any integer ...
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### Sensitivity and Low-Degree Approximation under Non-Uniform Distribution

I am searching for generalizations of analysis of Boolean functions when the input strings are distributed according to a general non-uniform distribution, possibly with arbitrary dependencies between ...
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### $p$-biased two-function hypercontractivity

The Hypercontractivity theorem (or Bonami Beckner inequality) is a very useful tool. Unfortunately, it isn't easy to carry over to other spaces than the uniform boolean cube. In Ryan O'Donnel's ...
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### What would faster Fourier Transform(FFT?) and/or multiplication algorithms imply?

There are many problems which have implications on P vs. NP and other complexity classes. Supposing that we're interested in Fourier transforms and/or multiplication algorithms, do faster Fourier ...
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### On FFT and trigonometric matrix eigenvalues

Let $N=2^n$ for a natural number $n$ and $B$ be the $N\times N$ square matrix of $0$'s and $1$'s  B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots ...
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### Is the nonnegativeness of a polynomial hard for $\mathsf{NP}_\mathbb{R}$?

It is clear that the following problem is in $\mathsf{NP}_\mathbb{R}$. Input: a list $P$ of triplets $(a,s,t)$ where $s$ and $t$ are nonnegative integers. Output: is there an $x\in \mathbb{R}$ such ...
### Is there a relation between $l_p$-norms of functions with same Fourier spectra but w.r.t different measures on the Hamming cube?
Informally, I want to ask if two functions $f$ and $g$ on the Hamming cube have the same Fourier spectra but w.r.t different measure and basis, then is $||f||_p$ related to $||g||_p$? (Where each $p$-...