Related papers: FKN Theorem on the biased cube

We investigate the influences of variables on a Boolean function $f$ based on the quantum Bernstein-Vazirani algorithm. A previous paper (Floess et al. in Math. Struct. in Comp. Science 23: 386, 2013) has proved that if a $n$-variable…

It is proved an amplification of Cusick-Cheon's conjecture on balanced Boolean functions in the cosets of the binary Reed-Muller code RM(k,m) of order k and length 2^m, in the cases where k = 1 or k >= (m-1)/2.

A remarkable discrete counterpart of the Gaussian function of one continuous variable can be defined by using a Jacobi theta function, that is, as the sum of a convergent series. We extend this approach to Gaussian functions of two…

Fourier series multiscale method, a concise and efficient analytical approach for multiscale computation, will be developed out of this series of papers. The second paper is concerned with simultaneous approximation to functions and their…

In a complete metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we study strict subsets, i.e. sets whose variational capacity with respect to a larger reference set is finite, in the case $p=1$.…

We prove sharp $L^2$ Fourier restriction inequalities for compact, smooth surfaces in $\mathbb{R}^3$ equipped with the affine surface measure or a power thereof. The results are valid for all smooth surfaces and the bounds are uniform for…

This manuscript includes some classical results we select apart from the new results we've found on the Analysis of Boolean Functions and Fourier-Entropy-Influence conjecture. We try to ensure the self-completeness of this work so that…

We present new algorithms to compute fundamental properties of a Boolean function given in truth-table form. Specifically, we give an O(N^2.322 log N) algorithm for block sensitivity, an O(N^1.585 log N) algorithm for `tree decomposition,'…

Due to a theorem by Orlov every exact fully faithful functor between the bounded derived categories of coherent sheaves on smooth projective varieties is of Fourier-Mukai type. We extend this result to the case of bounded derived categories…

New sufficient conditions for representation of a function via the absolutely convergent Fourier integral are obtained in the paper. In the main result, Theorem 1.1, this is controlled by the behavior near infinity of both the function and…

We derive a simple and precise approximation to probability density functions in sampling distributions based on the Fourier cosine series. After clarifying the required conditions, we illustrate the approximation on two examples: the…

The vector-valued $T(1)$ theorem due to Figiel, and a certain square function estimate of Bourgain for translations of functions with a limited frequency spectrum, are two cornerstones of harmonic analysis in UMD spaces. In this paper, a…

Let $V$ be a set of isolated points in $\RR^d$. Define a linear functional $\CL$ on the space of real polynomials restricted on $V$, $\CL f = \sum_{x \in V} f(x)\rho(x)$, where $\rho$ is a nonzero function on $V$. Polynomial subspaces that…

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…

We establish a sharp adjoint Fourier restriction inequality for the end-point Tomas-Stein restriction theorem on the circle under a certain arithmetic constraint on the support set of the Fourier coefficients of the given function. Such…

The Fourier-Entropy Influence (FEI) Conjecture states that for any Boolean function $f:\{+1,-1\}^n \to \{+1,-1\}$, the Fourier entropy of $f$ is at most its influence up to a universal constant factor. While the FEI conjecture has been…

We prove that the Fourier dimension of any Boolean function with Fourier sparsity $s$ is at most $O\left(s^{2/3}\right)$. Our proof method yields an improved bound of $\widetilde{O}(\sqrt{s})$ assuming a conjecture of…

Let $0 < \epsilon < 1/2$ be a noise parameter, and let $T_{\epsilon}$ be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a nonnegative function on $\{0,1\}^n$. We upper bound the entropy of $T_{\epsilon}…

Suppose $X$ is a uniformly distributed $n$-dimensional binary vector and $Y$ is obtained by passing $X$ through a binary symmetric channel with crossover probability $\alpha$. A recent conjecture by Courtade and Kumar postulates that…

In this paper we propose an approximation method for high-dimensional $1$-periodic functions based on the multivariate ANOVA decomposition. We provide an analysis on the classical ANOVA decomposition on the torus and prove some important…