Yeli Niu | Searcharxiv

arXiv2026

A Bloch type space associated with {\lambda}-analytic functions

For $\lambda\ge0$, the so-called $\lambda$-analytic functions are defined in terms of the (complex) Dunkl operators $D_{z}$ and $D_{\bar{z}}$. In the paper we introduce a Bloch type space on the disk ${\mathbb D}$ associated with $\lambda$-analytic functions, called the $\lambda$-Bloch space and denoted by ${\mathfrak{B}}_{\lambda}({\mathbb D})$. Various properties of the $\lambda$-Bloch space ${\mathfrak{B}}_{\lambda}({\mathbb D})$ are proved. We give a characterization of functions in ${\mathfrak{B}}_{\lambda}({\mathbb D})$ by means of the higher-order operators $(D_z\circ z)^n$ for $n\ge2$. A general integral operator is proved to be bounded from $L^{\infty}({\mathbb D})$ onto ${\mathfrak{B}}_{\lambda}({\mathbb D})$, and as an application, the dual relation of ${\mathfrak{B}}_{\lambda}({\mathbb D})$ and the $\lambda$-Bergman space ($p=1$) is verified.

math.CV↗

arXiv2020

Discretization on high-dimensional domains

Let $\mu$ be a Borel probability measure on a compact path-connected metric space $(X, \rho)$ for which there exist constants $c,\beta>1$ such that $\mu(B) \geq c r^{\beta}$ for every open ball $B\subset X$ of radius $r>0$. For a class of Lipschitz functions $\Phi:[0,\infty)\to R$ that piecewisely lie in a finite-dimensional subspace of continuous functions, we prove under certain mild conditions on the metric $\rho$ and the measure $\mu$ that for each positive integer $N\geq 2$, and each $g\in L^\infty(X, d\mu)$ with $\|g\|_\infty=1$, there exist points $y_1, \ldots, y_{ N}\in X$ and real numbers $\lambda_1, \ldots, \lambda_{ N}$ such that for any $x\in X$, \begin{align*} & \left| \int_X \Phi (\rho (x, y)) g(y) \,d \mu (y) - \sum_{j = 1}^{ N} \lambda_j \Phi (\rho (x, y_j)) \right| \leq C N^{- \frac{1}{2} - \frac{3}{2\beta}} \sqrt{\log N}, \end{align*} where the constant $C>0$ is independent of $N$ and $g$. In the case when $X$ is the unit sphere $S^d$ of $R^{d+1}$ with the ususal geodesic distance, we also prove that the constant $C$ here is independent of the dimension $d$. Our estimates are better than those obtained from the standard Monte Carlo methods, which typically yield a weaker upper bound $N^{-\frac12}\sqrt{\log N}$.

math.NA↗