Related papers: The H\"{o}rmander--Bernhardsson extremal function

For an appropriate class of convex functions $\phi$, we study the Fourier extension operator on the surface $\{(y, |y|^2+\phi(y)):y\in\mathbb{R}^2\}$ equipped with projection measure. For the corresponding extension inequality, we compute…

Let $A$ and $B$ be almost commuting (i.e., the commutator $AB-BA$ belongs to trace class) self-adjoint operators. We construct a functional calculus $\varphi\mapsto\varphi(A,B)$ for functions $\varphi$ in the Besov class…

We prove Conjecture~2 of Bondarenko, Ortega-Cerd\`a, Radchenko, and Seip for the three-term recurrence attached to the H\"ormander--Bernhardsson extremal function $\varphi$. More precisely, define \[ \widetilde u_{-1}=0,\qquad \widetilde…

Let denote by $S(\phi)$ the extremal operator defined by the compression of the unilateral shift $S$ to the model subspace $ H(\phi)=H^{2} \ominus \phi H^{2} $ as the following $S(\phi)f(z)=P(zf(z)),$ where $P$ denotes the orthogonal…

We prove that there exists an extremal function to the Airy Strichartz inequality, $e^{-t\partial_x^3}: L^2(\mathbb{R})\to L^8_{t,x}(\mathbb{R}^2)$ by using the linear profile decomposition. Furthermore we show that, if $f$ is an…

In the present paper, we characterize the nonnegative functions $\varphi$ for which the multi-parameter Hausdorff operator $\mathcal H_\varphi$ generated by $\varphi$ is bounded on the multi-parameter Hardy space $H^1(\mathbb…

We prove that every measurable function $f:\,[0,a]\to\mathbb{C}$ such that $|f|=1$ a.e. on $[0,a]$ is an extreme point of the unit ball of the Lorentz space $\Lambda(\varphi)$ on $[0,a]$ whenever $\varphi$ is a not linear, strictly…

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…

We study in this paper very badly approximable matrix functions on the unit circle $\T$, i.e., matrix functions $\Phi$ such that the zero function is a superoptimal approximation of $\Phi$. The purpose of this paper is to obtain a…

For a real analytic periodic function $\phi:\mathbb{R}\to\mathbb{R}^d$, an integer $b \ge 2$ and $\lambda\in(1/b,1)$, we prove that the box dimension and the Hausdorff dimension of the graph of the Weierstrass function…

We establish the asymptotic theory of least absolute deviation estimators for AR(1) processes with autoregressive parameter satisfying $n(\rho_n-1)\to\gamma$ for some fixed $\gamma$ as $n\to\infty$, which is parallel to the results of…

In this paper we improve Bernoulli comparison. The result works for independent Rademacher random variables $(\varepsilon_i)_{i\geq1}$ and states that we can compare $\mathbb{E}\sup_{t\in T}\sum_{i\geq1}\varphi_{i}(t)\varepsilon_i$ with…

Let $H$ be the symmetric second-order differential operator on $L_2(\Ri)$ with domain $C_c^\infty(\Ri)$ and action $H\varphi=-(c \varphi')'$ where $ c\in W^{1,2}_{\rm loc}(\Ri)$ is a real function which is strictly positive on…

In this paper we prove that if $X $ is a Banach space, then for every lower semi-continuous bounded below function $f, $ there exists a $\left(\varphi_1, \varphi_2\right)-$convex function $g, $ with arbitrarily small norm, such that $f + g…

We extend Hochman's work on exponentially separated self-similar measures on $\mathbb{R}$ to the real analytic setting. More precisely, let $\Phi=\left\{ \varphi_{i}\right\} _{i\in\Lambda}$ be an iterated function system on $I:=[0,1]$…

Based on elementary methods and techniques, the explicit formula for the generalized Euler function $\varphi_{e}(n)(e=8,12)$ is given, and then a sufficient and necessary condition for $\varphi_{8}(n)$ or $\varphi_{12}(n)$ to be odd is…

The present paper considers a problem of estimating a linear functional $\Phi=\int_{-\infty}^\infty \varphi(x) f(x)dx$ of an unknown deconvolution density $f$ on the basis of i.i.d. observations $Y_i = \theta_i + \xi_i$ where $\xi_i$ has a…

Extending earlier work by Rogers, Wiener-Hopf factorisation is studied for a class of functions closely related to Nevanlinna-Pick functions and complete Bernstein functions. The name 'Rogers functions' is proposed for this class. Under…

A real valued function $f$ defined on a real open interval $I$ is called $\Phi$-monotone if, for all $x,y\in I$ with $x\leq y$ it satisfies $$ f(x)\leq f(y)+\Phi(y-x), $$ where $\Phi:[0,\ell(I)[\,\to\mathbb{R}_+$ is a given nonnegative…

The generalised Gegenbauer functions of fractional degree (GGF-Fs), denoted by ${}^{r\!}G^{(\lambda)}_\nu(x)$ (right GGF-Fs) and ${}^{l}G^{(\lambda)}_\nu(x)$ (left GGF-Fs) with $x\in (-1,1),$ $\lambda>-1/2$ and real $\nu\ge 0,$ are special…