Related papers: On approximately monotone and approximately H\"old…

In this paper, our focus lies on the study of the second-order variational analysis of orthogonally invariant matrix functions. It is well-known that an orthogonally invariant matrix function is an extended-real-value function defined on…

We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {-1,1}^n. This result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the…

This is a conitunation of [1] and [2]. We prove that if function $f$ belongs to the class $\Lambda_{\omega} \overset{\text{def}}{=} \{f: \omega_{f}(\delta)\leq \text{const} \omega(\delta)\} $ for an arbitrary modulus of continuity $\omega$,…

The approximate degree of a Boolean function $f: \{-1, 1\}^n \to \{-1, 1\}$ is the minimum degree of a real polynomial that approximates $f$ to within error $1/3$ in the $\ell_\infty$ norm. In an influential result, Aaronson and Shi (J. ACM…

Suppose that f is a boolean function from F_2^n to {0,1} with spectral norm (that is the sum of the absolute values of its Fourier coefficients) at most M. We show that f may be expressed as +/- 1 combination of at most 2^(2^(O(M^4)))…

The notion off-ideals is recent and has been studied in the papers[1] [2], [5], [10], [11], [12], [13], [14] and [15]. In this paper, we have generalized the idea off-ideals to quasi f-ideals. This extended class of ideals is much bigger…

The Fourier Entropy-Influence (FEI) Conjecture of Friedgut and Kalai states that ${\bf H}[f] \leq C \cdot {\bf I}[f]$ holds for every Boolean function $f$, where ${\bf H}[f]$ denotes the spectral entropy of $f$, ${\bf I}[f]$ is its total…

Let $\Omega_1,\Omega_2$ be two disjoint open sets in $\mathbf C^n$ whose boundaries share a smooth real hypersurface $M$ as relatively open subsets. Assume that $\Omega_i$ is equipped with a complex structure $J^i$ which is smooth up to…

The main aim of this paper is to study the functional inequality \begin{equation*} \int_{[0,1]}f\bigl((1-t)x+ty\bigr)d\mu(t)\geq 0, \qquad x,y\in I \mbox{ with } x<y, \end{equation*} for a continuous unknown function $f:I\to{\mathbb R}$,…

The Fourier extension method, also known as the Fourier continuation method, is a method for approximating non-periodic functions on an interval using truncated Fourier series with period larger than the interval on which the function is…

We show a procedure that, given oracle access to a function $f\colon \{0,1\}^n\to\{0,1\}$, produces oracle access to a function $f'\colon \{0,1\}^{n'}\to\{0,1\}$ such that if $f$ is monotone, then $f'$ is monotone, and if $f$ is…

We study function estimation in the empirical Bayes setting for Poisson and normal means. Specifically, given observations $Y_i\sim f(\cdot; \theta_i)$ with latent parameters $\theta_i\sim \pi$, the goal is to estimate…

We characterize the function $\varphi$ of minimal $L^1$ norm among all functions $f$ of exponential type at most $\pi$ for which $f(0)=1$. This function, studied by H\"{o}rmander and Bernhardsson in 1993, has only real zeros $\pm \tau_n$,…

Let $\{f_i:\mathbb{F}_p^i \to \{0,1\}\}$ be a sequence of functions, where $p$ is a fixed prime and $\mathbb{F}_p$ is the finite field of order $p$. The limit of the sequence can be syntactically defined using the notion of ultralimit.…

Given an anisotropic integrand $F:\text{Gr}_k(\mathbb R^n)\to(0,\infty)$, we can generalize the classical isotropic area by looking at the functional $$\mathcal{F}(\Sigma^k):=\int_\Sigma F(T_x\Sigma)\,d\mathcal{H}^k.$$ While a monotonicity…

In this paper we consider a fragment of the first-order theory of the real numbers that includes systems of equations of continuous functions in bounded domains, and for which all functions are computable in the sense that it is possible to…

A real arithmetic function f is multiplicatively monotonous if f (mn) -- f (m) has constant sign for m, n positive integers. Properties and examples of such functions are discussed, with applications to positive hermitian…

We construct a H\"older continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We say that a function with…

In this paper we study optimal estimates for two functionals involving the anisotropic $p$-torsional rigidity $T_p(\Omega)$, $1<p<+\infty$. More precisely, we study $\Phi(\Omega)=\frac{T_p(\Omega)}{|\Omega|M(\Omega)}$ and…

For a real-valued function $f$ on a metric measure space $(X,d,\mu)$ the Hardy-Littlewood maximal-function of $f$ is given by the following `supremum-norm':…