Related papers: Boolean functions whose Fourier transform is conce…
Chang's lemma (Duke Mathematical Journal, 2002) is a classical result with applications across several areas in mathematics and computer science. For a Boolean function $f$ that takes values in {-1,1} let $r(f)$ denote its Fourier rank. For…
We investigate the approximability of several classes of real-valued functions by functions of a small number of variables ({\em juntas}). Our main results are tight bounds on the number of variables required to approximate a function…
Let $A$ be a subset of a finite abelian group such that $A$ has a small difference set $A-A$ and the density of $A$ is small. We prove that, counter--intuitively, the smallness (in terms of $|A-A|$) of the Fourier coefficients of $A$…
This paper considers the Fourier transform over the slice of the Boolean hypercube. We prove a relationship between the Fourier coefficients of a function over the slice, and the Fourier coefficients of its restrictions. As an application,…
The discrete Fourier transform of the greatest common divisor is a multiplicative function that generalises both the gcd-sum function and Euler's totient function. On the one hand it is the Dirichlet convolution of the identity with…
We study the $\ell^1$-summability of functions in the $d$-dimensional torus $\mathbb{T}^d$ and so-called $\ell^1$-invariant functions. Those are functions on the torus whose Fourier coefficients depend only on the $\ell^1$-norm of their…
We prove a Chernoff-like large deviation bound on the sum of non-independent random variables that have the following dependence structure. The variables $Y_1,...,Y_r$ are arbitrary Boolean functions of independent random variables…
We introduce an algorithm to compute the functions belonging to a suitable set ${\mathscr F}$ defined as follows: $f\in {\mathscr F}$ means that $f(s,x)$, $s\in A\subset {\mathbb R}$ being fixed and $x>0$, has a power series expansion…
Let F(R^n) be the algebra of Fourier transforms of functions from L_1(R^n), K(R^n) be the algebra of Fourier transforms of bounded complex Borel measures in R^n and W be Wiener algebra of continuous 2pi-periodic functions with absolutely…
A deep approximation is an approximating function defined by composing more than one layer of simple functions. We study deep approximations of functions of one variable using layers consisting of low-degree polynomials or simple conformal…
This paper argues that the ideas underlying the renormalization group technique used to characterize phase transitions in condensed matter systems could be useful for distinguishing computational complexity classes. The paper presents a…
A seminal result of Nisan and Szegedy (STOC, 1992) shows that for any total Boolean function, the degree of the real polynomial that computes the function, and the minimal degree of a real polynomial that point-wise approximates the…
The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory. For example, symmetric functions, that is, functions that are invariant under the action of $S_n$, is an important class…
The main goal is to interpret the Askey-Wilson function and the corresponding transform pair on the quantum SU(1,1) group. A weight on the C^*-algebra of continuous functions vanishing at infinity on the quantum SU(1,1) group is studied,…
We study reduction schemes for functions of "many" variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit…
We generalize the celebrated isoperimetric inequality of Khot, Minzer, and Safra~(SICOMP 2018) for Boolean functions to the case of real-valued functions $f \colon \{0,1\}^d\to\mathbb{R}$. Our main tool in the proof of the generalized…
The largest Hamming distance between a Boolean function in $n$ variables and the set of all affine Boolean functions in $n$ variables is known as the covering radius $\rho_n$ of the $[2^n,n+1]$ Reed-Muller code. This number determines how…
In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number…
In an effort to extend classical Fourier theory, Hedenmalm and Montes-Rodr\'{\i}guez (2011) found that the function system \[ e_m(x)=e^{i\pi mx},\quad e_n^\dagger(x)=e_n(-1/x)=e^{-i\pi n/x} \] is weak-star complete in…
We consider certain finite sets of circle-valued functions defined on intervals of real numbers and estimate how large the intervals must be for the values of these functions to be uniformly distributed in an approximate way. This is used…