Related papers: On existence of Becker extension

Let S be a symmetric relation with deficiency index (1,1). In this article, we extend Krein`s resolvent formalism in order to describe all, not necessarily self-adjoint, extensions $S \subset \tilde{A}$ with $\varrho(\tilde{A})\neq…

We investigate the best constant $J(n,d)$ such that Jackson's inequality \[ \inf_{\mathrm{deg}(g) \leq d} \|f - g\|_{\infty} \leq J(n,d) \, s(f), \] holds for all functions $f$ on the hypercube $\{0,1\}^n$, where $s(f)$ denotes the…

For two graphs $F,H$ and a positive integer $n$, the function $f_{F,H}(n)$ denotes the largest $m$ such that every $H$-free graph on $n$ vertices contains an $F$-free induced subgraph on $m$ vertices. This function has been extensively…

In this paper, we consider the following singular integral \begin{equation*} T_jf(x)=K_j*f(x), K_j(x)=\frac{x_j}{|x|^{n+1-\beta}}, \end{equation*} where $x\in R^n, 0\le \beta<n, j=1,2,\cdots, n$. When $\beta=0$, it corresponds to the Riesz…

Given a real number $x>0$, we determine $q_s(x):=\inf\mathscr{U}(x)$, where $\mathscr{U}(x)$ is the set of all bases $q\in(1,2]$ for which $x$ has a unique expansion of $0$'s and $1$'s. We give an explicit description of $q_s(x)$ for…

Let $E \subset \mathbb{R}^n$ be a compact set, and $f:E \to \mathbb{R}$. How can we tell if there exists a convex extension $F \in C^{1,1}(\mathbb{R}^n)$ of $f$, i.e. satisfying $F|_E = f|_E$? Assuming such an extension exists, how small…

A $k$-regular partition into distinct parts is a partition into distinct parts with no part divisible by $k$. In this paper, we provide a general method to establish the unimodality of $k$-regular partition into distinct parts where the…

Let $r\in\mathbb{C}$, let $K$ be a finite extension of $\mathbb{Q}$, let $I_K$ be the monoid of integral ideals in the ring of integers $\mathcal{O}_K$ of $K$, and let $\chi$ be a Dirichlet character. Then define the twisted ideal divisor…

In this note we prove that, for $p>0$, $L_{p}[0,1]\smallsetminus\bigcup_{q\in(p,\infty)}L_{q}[0,1]$ is $(\alpha,\mathfrak{c})$-spaceable if, and only if, $\alpha<\aleph_{0}$. Such a problem first appears in [V. F\'avaro, D. Pellegrino, D.…

Let ${\bf P}_k^{(\alpha, \beta)} (x)$ be an orthonormal Jacobi polynomial of degree $k.$ We will establish the following inequality \begin{equation*} \max_{x \in [\delta_{-1},\delta_1]}\sqrt{(x- \delta_{-1})(\delta_1-x)}…

We consider a Neumann problem for strictly convex variational functionals of linear growth. We establish the existence of minimisers among $\operatorname{W}^{1,1}$-functions provided that the domain under consideration is simply connected.…

We prove that if $\Omega\subset \mathbb R^n$ is a bounded open set and $n\alpha> {\rm dim}_b (\partial \Omega) = d$, then the Brouwer degree deg$(v,\Omega,\cdot)$ of any H\"older function $v\in C^{0,\alpha}\left (\Omega, \mathbb…

We prove that, given a constant $K> 2$ and a bounded linear operator $T$ from a JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one functional $\psi\in E^*$ satisfying $$\|T(x)\| \leq K \, \|T\| \, \|x\|_{\psi},$$…

Motivated by the Maximum Theorem for convex functions (in the setting of linear spaces) and for subadditive functions (in the setting of Abelian semigroups), we establish a Maximum Theorem for the class of generalized convex functions,…

Various kinds of Stieltjes integrals using gauge integration have become highly popular in the field of differential equations and other applications. In the theories of integration and of ordinary differential equations, convergence…

Let $f(x)$ be a real function which has $(n+1)$-th derivative on an interval $[a, b]$. For any point $x_0\in (a, b)$ and any integer $0\leq k\leq n$, denote by $S_{k,x_0}(x)$ the $k$-th truncation of the Taylor expansion of $f(x)$ at $x_0$,…

Given $s\in(1,2]$, define $$H_s[0,1]=\{f\in C[0,1]:{\dim}_HG_f([0,1])=s\}$$ and $$\overline{B}_s[0,1]=\{f\in C[0,1]:\overline{{\dim}}_BG_f([0,1])=s\}.$$ The main goal of this paper is to study the $(\alpha,\beta)$-lineability/spaceability…

Let $q\in(1,2)$. A $q$-expansion of a number $x$ in $[0,\frac{1}{q-1}]$ is a sequence $(\delta_i)_{i=1}^\infty\in\{0,1\}^{\mathbb{N}}$ satisfying $$ x=\sum_{i=1}^\infty\frac{\delta_i}{q^i}.$$ Let $\mathcal{B}_{\aleph_0}$ denote the set of…

A remarkable theorem of Besicovitch is that an integrable function $f$ on $\mathbb{R}^2$ is strongly differentiable if and only if its associated strong maximal function $M_S f$ is finite a.e. We provide an analogue of Besicovitch's result…

Let $R_j$ denote the $j^{\text{th}}$ Riesz transform on $\mathbb{R}^n$. We prove that there exists an absolute constant $C>0$ such that \begin{align*} |\{|R_jf|>\lambda\}|\leq C\left(\frac{1}{\lambda}\|f\|_{L^1(\mathbb{R}^n)}+\sup_{\nu}…