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

A finite Borel measure $\mu$ in ${\mathbb R}^d$ is called a frame-spectralmeasure if it admits an exponential frame (or Fourier frame) for $L^2(\mu)$. Ithas been conjectured that a frame-spectral measure must be translationallyabsolutely continuous, which is a criterion describing the local uniformity ofa measure on its support. In this paper, we show that if any measures $\nu$ and$\lambda$ without atoms whose supports form a packing pair, then $\nu\ast\lambda +\delta_t\ast\nu$ is translationally singular and it does not admit anyFourier frame. In particular, we show that the sum of one-fourth andone-sixteenth Cantor measure $\mu_4+\mu_{16}$ does not admit any Fourier frame.We also interpolate the mixed-type frame-spectral measures studied by Lev andthe measure we studied. In doing so, we demonstrate a discontinuity behavior:For any anticlockwise rotation mapping $R_{\theta}$ with $\theta\ne \pm\pi/2$,the two-dimensional measure $\rho_{\theta} (\cdot): =(\mu_4\times\delta_0)(\cdot)+(\delta_0\times\mu_{16})(R_{\theta}^{-1}\cdot)$,supported on the union of $x$-axis and $y=(\cot \theta)x$, always admit aFourier frame. Furthermore, we can find $\{e^{2\pi i\langle\lambda,x\rangle}\}_{\lambda\in\Lambda_{\theta}}$ such that it forms aFourier frame for $\rho_{\theta}$ with frame bounds independent of $\theta$.Nonetheless, $\rho_{\pm\pi/2}$ does not admit any Fourier frame.

Translational absolute continuity and Fourier frames on a sum of singular measures