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

We investigate the rate of convergence of linear sampling numbers of theembedding $H^{\alpha,\beta} (\mathbb{T}^d) \hookrightarrow H^\gamma(\mathbb{T}^d)$. Here $\alpha$ governs the mixed smoothness and $\beta$ theisotropic smoothness in the space $H^{\alpha,\beta}(\mathbb{T}^d)$ of hybridsmoothness, whereas $H^{\gamma}(\mathbb{T}^d)$ denotes the isotropic Sobolevspace. If $\gamma>\beta$ we obtain sharp polynomial decay rates for the firstembedding realized by sampling operators based on "energy-norm based sparsegrids" for the classical trigonometric interpolation. This complements earlierwork by Griebel, Knapek and D\~ung, Ullrich, where general linearapproximations have been considered. In addition, we study the embedding$H^\alpha_{mix} (\mathbb{T}^d) \hookrightarrow H^{\gamma}_{mix}(\mathbb{T}^d)$and achieve optimality for Smolyak's algorithm applied to the classicaltrigonometric interpolation. This can be applied to investigate the samplingnumbers for the embedding $H^\alpha_{mix} (\mathbb{T}^d) \hookrightarrowL_q(\mathbb{T}^d)$ for $2<q\leq \infty$ where again Smolyak's algorithm yieldsthe optimal order. The precise decay rates for the sampling numbers in thementioned situations always coincide with those for the approximation numbers,except probably in the limiting situation $\beta = \gamma$ (including theembedding into $L_2(\mathbb{T}^d)$). The best what we could prove there is a(probably) non-sharp results with a logarithmic gap between lower and upperbound.

Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in $H^\gamma$