English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

If $x_1,\dots,x_m$ are finitely many points in $\mathbb{R}^d$, let$E_\epsilon=\cup_{i=1}^m\,x_i+Q_\epsilon$, where $Q_\epsilon=\{x\in\mathbb{R}^d,\,\,|x_i|\le \epsilon/2, \, i=1,...,d\}$ and let $\hat f$ denotethe Fourier transform of $f$. Given a positive Borel measure $\mu$ on$\mathbb{R}^d$, we provide a necessary and sufficient condition for the frameinequalities $$ A\,\|f\|^2_2\le \int_{\mathbb{R}^d}\,|\hatf(\xi)|^2\,d\mu(\xi)\le B\,\|f\|^2_2,\quad f\in L^2(E_\epsilon), $$ to hold forsome $A,B>0$ and for some $\epsilon>0$ sufficiently small. If $m=1$, we showthat the limits of the optimal lower and upper frame bounds as$\epsilon\rightarrow 0$ are equal, respectively, to the lower and upperBeurling density of $\mu$. When $m>1$, we extend this result by defining amatrix version of Beurling density. Given a (possibly dense) subgroup $G$ of$\mathbb{R}$, we then consider the problem of characterizing those measures$\mu$ for which the inequalities above hold whenever $x_1,\dots,x_m$ arefinitely many points in $G$ (with $\epsilon$ depending on those points, but not$A$ or $B$). We point out an interesting connection between this problem andthe notion of well-distributed sequence when $G=a\,\mathbb{Z}$ for some $a>0$.Finally, we show the existence of a discrete set $\Lambda$ such that themeasure $\mu=\sum_{\lambda}\,\delta_\lambda$ satisfy the property above for thewhole group $\mathbb{R}$.

Beurling densities and frames of exponentials on the union of small balls