Related papers: Repeated integrals of increasing functions

The noise sensitivity of a Boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ is one of its fundamental properties. A function of a positive noise parameter $\delta$, it is denoted as $NS_{\delta}[f]$. Here we study the algorithmic problem…

We investigate the convexity property on $(0,1)$ of the functions $\varphi_{a,b,c}$ and $1/\varphi_{a,b,c}$, where $$\varphi_{a,b,c}(x)= \frac{c-\log(1-x)}{\,_2F_1(a,b,a+b,x)},$$ whenever $a,b\geq 0$ and $a+b\leq 1$. We Show that…

It is not surprising that one should expect that the degree of constrained (shape preserving) approximation be worse than the degree of unconstrained approximation. However, it turns out that, in certain cases, these degrees are the same.…

Let $b\geq2$ be an integer and $A=(a_{n})_{n=1}^{\infty}$ be a strictly increasing subsequence of positive integers with $\eta:=\limsup\limits_{n\to\infty}\frac{a_{n+1}}{a_{n}}<+\infty$. For each irrational real number $\xi$, we denote by…

We focus on a sequence of functions $\{f_n\}$, defined on a compact manifold with boundary $S$, converging in the $C^k$ metric to a limit $f$. A common assumption implicitly made in the empirical sciences is that when such functions…

We determine sufficient conditions under which certain recursively defined functions are well defined for all real inputs. Given a function $f:\mathbb R\to\mathbb R$, call a decreasing sequence $x_1>x_2>x_3>\cdots$ "$f$-bad" if…

Let $C$ be a subset of $\mathbb{R}^n$ (not necessarily convex), $f:C\to\mathbb{R}$ be a function, and $G:C\to\mathbb{R}^n$ be a uniformly continuous function, with modulus of continuity $\omega$. We provide a necessary and sufficient…

We enhance the approximation capabilities of algebraic polynomials by composing them with homeomorphisms. This composition yields families of functions that remain dense in the space of continuous functions, while enabling more accurate…

By using the complete discrimination system for polynomials, we study the number of positive solutions in {\small $C[0,1]$} to the integral equation {\small $\phi (x)=\int_0^1k(x,y)\phi ^n(y)dy$}, where {\small…

The paper deals with the problem of ideals of $H^\infty$: describe increasing functions $\phi\ge 0$ such that for all bounded analytic functions $f_1,f_2,...,f_n, \tau$ in the unit disc $D$ the condition $|\tau(z) | \le \phi(\sum_k…

Let $(b_k)_{k = 0}^\infty$ be strictly decreasing sequence of real numbers such that $b_0 = 1$ and $\{f_k:[b_k,b_{k-1}]\to [0,1]\}_{k\in\N}$ be decreasing functions such that $f_k(b_k) = 1$ and $f_k(b_{k-1}) = 0$, $k = 1, 2, \dots$. By…

Let $u\not\equiv -\infty$ be a subharmonic function on the complex plane $\mathbb C$. Then for any function $r\colon\mathbb C\to (0,1]$ satisfying the condition $$\inf_{z\in\mathbb C}\frac{\ln r(z)}{\ln(2+|z|)}>-\infty,$$ there is an entire…

The model of relative-error property testing of Boolean functions has been the subject of significant recent research effort [CDH+24][CPPS25a][CPPS25b] In this paper we consider the problem of relative-error testing an unknown and arbitrary…

Kurokawa and Koyama's multiple cosine function $\mathcal{C}_{r}(x)$ and Kurokawa's multiple sine function $S_{r}(x)$ are generalizations of the classical cosine and sine functions from their infinite product representations, respectively.…

In this note we introduce the notion of Newton--C\^{o}tes functionals corrected by L\'{e}vy areas, which enables us to consider integrals of the type $\int f(y) \mathrm{d}x,$ where $f$ is a ${\mathscr{C}}^{2m}$ function and $x,y$ are real…

If $f(x,k)$ is the Jost solution and $f(x) = f(0,k)$, then the $I$-function is $I(k) := \frac{f^\prime(0,k)}{f(k)}$. It is proved that $I(k)$ is in one-to-one correspondence with the scattering triple ${\mathcal S} :=\{S(k), k_j, s_j, \quad…

This article characterizes the associativity of two-place functions $T: [0,1]^2\rightarrow [0,1]$ defined by $T(x,y)=f^{(-1)}(F(f(x),f(y)))$ where $F:[0,1]^2\rightarrow[0,1]$ is a triangular norm (even a triangular subnorm), $f:…

We obtain results for the following question where $m\ge 1$ and $n\ge 2$ are integers. {\bf Question.} For which continuous functions $f\colon [0,\infty)\to [0,\infty)$ does there exist a continuous function $\phi\colon (0,1)\to (0,\infty)$…

Let $\alpha$ and $\beta$ be irrational real numbers and $0<\F<1/30$. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy $\|q\alpha\|\cdot\|q\beta\|<\F$. If we choose $\F$ as a function of $Q$ we get…

Let $\mathcal{S}$ denote the family of all functions that are analytic and univalent in the unit disk $\mathbb{D}:=\{z: |z|<1\}$ and satisfy $f(0)=f^{\prime}(0)-1=0$. In the present paper, we consider certain subclasses of univalent…