Related papers: Range of certain convolution operators and reconst…

Given a low-frequency sample of the infinitely divisible moving average random field $\{\int_{\mathbb{R}^d}f(t-x)\Lambda (dx), t\in \mathbb{R}^d\}$, in [13] we proposed an estimator $\hat{uv_0}$ for the function $\mathbb{R}\ni x\mapsto…

We investigate the conditions under which the space of bounded harmonic functions of a probability measure $\mu$ on a group $G$ is contained in that of another measure $\theta$. We establish that asymptotic commutativity, defined by the…

For artificial deep neural networks, we prove expression rates for analytic functions $f:\mathbb{R}^d\to\mathbb{R}$ in the norm of $L^2(\mathbb{R}^d,\gamma_d)$ where $d\in {\mathbb{N}}\cup\{ \infty \}$. Here $\gamma_d$ denotes the Gaussian…

Let $X$ be a locally compact Polish space. Let $\mathbb K(X)$ denote the space of discrete Radon measures on $X$. Let $\mu$ be a completely random discrete measure on $X$, i.e., $\mu$ is (the distribution of) a completely random measure on…

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains $C$, with non-smooth boundary, in possibly non-compact manifolds. Assuming $C$ is a submanifold with corners, or…

If $\mu$ is a finite measure on the unit disc and $k\ge 0$ is an integer, we study a generalization derived from Englis's work, $T_\mu^{(k)}$, of the traditional Toeplitz operators on the Bergman space $A^2$, which are the case $k=0$. Among…

For $1 \le t < \infty ,$ a compact subset $K$ of the complex plane $\mathbb C,$ and a finite positive measure $\mu$ supported on $K,$ $R^t(K, \mu)$ denotes the closure in $L^t (\mu )$ of rational functions with poles off $K.$ The paper…

We prove a Lusin approximation of functions of bounded variation. If $f$ is a function of bounded variation on an open set $\Omega\subset X$, where $X=(X,d,\mu)$ is a given complete doubling metric measure space supporting a $1$-Poincar\'e…

In this article we show the following result: if $C$ is an $n$-dimensional convex and compact subset, $f:C\rightarrow[0,\infty)$ is concave, and $\phi:[0,\infty)\rightarrow[0,\infty)$ is a convex function with $\phi(0)=0$, we then…

We use the principle of almost orthogonality to give a new and simple proof that a sparse Lerner operator is bounded on a matrix- or operator-weighted space $L_W^{2}(\mu)$, where $\mu$ is a doubling measure on $\R^d$ if and only if the…

We study $L^p(\mu) \to L^q(\nu)$ mapping properties of the convolution operator $ T_{\lambda}f(x)=\lambda*(f\mu)(x)$ and of the corresponding maximal operator $ {\mathcal T}_{\lambda}f(x)=\sup_{t>0} |\lambda_t*(f\mu)(x)|$, where $\lambda$…

In this paper, we construct a class of random measures $\mu^{\mathbf{n}}$ by infinite convolutions. Given infinitely many admissible pairs $\{(N_{k}, B_{k})\}_{k=1}^{\infty}$ and a positive integral sequence…

We consider iterated function systems (IFS) in ${\mathbb R}^d$ for $d\ge 3$ of the form $\{f_j(x) = \lambda {\mathcal O} x + a_j\}_{j=0}^m$, with $a_0=0$ and $m\ge 1$. Here $\lambda\in (0,1)$ is the contraction ratio and ${\mathcal O}$ is…

Given a bounded symmetric domain $D$ in $\mathbb C^n$, we consider the Clark measures $\mu_\alpha$, $\alpha\in \mathbb T$, associated with a rational inner function $\varphi$ from $D$ into the unit disc in $\mathbb C$. We show that…

We show that $C(X)$ admits an equivalent pointwise lower semicontinuous locally uniformly rotund norm provided $X$ is Fedorchuk compact of spectral height 3. In other words $X$ admits a fully closed map $f$ onto a metric compact $Y$ such…

We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using…

Let $\mu$ be a compactly supported absolutely continuous probability measure on ${\Bbb R}^n$, we show that $\mu$ admits Fourier frames if and only if its Radon-Nikodym derivative is upper and lower bounded almost everywhere on its support.…

We show a general phenomenon of the constrained functional value for densities satisfying general convexity conditions, which generalizes the observation in Bobkov and Madiman (2011) that the entropy per coordinate in a log-concave random…

Let $\mu$ be a positive Borel measure on the interval $[0,1)$. For $\alpha>0$, the generalized Hankel matrix $\mathcal{H}_{\mu, \alpha}=(\mu_{n, k, \alpha})_{n, k \geq 0}$ with entries $\mu_{n, k, \alpha}=\int_{[0,1)}…

We consider C^2 families t->f_t of C^4 nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure m_t of f_t, as a function of t on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming…