Sampling on energy-norm based sparse grids for the optimal recovery of Sobolev type functions in

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• 作者：Glenn Byrenheid^a ; Dinh Dũng^b ; ^{dinhzung@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Winfried Sickel^c ; Tino Ullrich^a
• 刊名：Journal of Approximation Theory
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：207
• 期：Complete
• 页码：207-231
• 全文大小：297 K

文摘

We investigate the rate of convergence of linear sampling numbers of the embedding d="mmlsi4" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516000356&_mathId=si4.gif&_user=111111111&_pii=S0021904516000356&_rdoc=1&_issn=00219045&md5=298a495757dda77a6eed65fae09eef4c" title="Click to view the MathML source">H^α,β(T^d)↪H^γ(T^d)dden">de">. Here d="mmlsi5" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516000356&_mathId=si5.gif&_user=111111111&_pii=S0021904516000356&_rdoc=1&_issn=00219045&md5=c1276ba209ac58c1e09935762a3e7bfc" title="Click to view the MathML source">αdden">de">$\alpha$ governs the mixed smoothness and d="mmlsi6" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516000356&_mathId=si6.gif&_user=111111111&_pii=S0021904516000356&_rdoc=1&_issn=00219045&md5=e2da10f9bed74b54cb0435af6f515685" title="Click to view the MathML source">βdden">de">$\beta$ the isotropic smoothness in the space d="mmlsi7" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516000356&_mathId=si7.gif&_user=111111111&_pii=S0021904516000356&_rdoc=1&_issn=00219045&md5=ec2f2479f9f4453ab39784983b5c738e" title="Click to view the MathML source">H^α,β(T^d)dden">de"> of hybrid smoothness, whereas d="mmlsi8" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516000356&_mathId=si8.gif&_user=111111111&_pii=S0021904516000356&_rdoc=1&_issn=00219045&md5=cd33e584050bea3326f51c31f0dd283a" title="Click to view the MathML source">H^γ(T^d)dden">de"> denotes the isotropic Sobolev space. If d="mmlsi9" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516000356&_mathId=si9.gif&_user=111111111&_pii=S0021904516000356&_rdoc=1&_issn=00219045&md5=692c254ee5fecdb517e5daaa90ba0a19" title="Click to view the MathML source">γ>βdden">de">$\gamma >\beta$ we obtain sharp polynomial decay rates for the first embedding realized by sampling operators based on &ldquo;energy-norm based sparse grids&rdquo; for the classical trigonometric interpolation. This complements earlier work by Griebel, Knapek and Dũng, Ullrich, where general linear approximations have been considered. In addition, we study the embedding d="mmlsi10" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516000356&_mathId=si10.gif&_user=111111111&_pii=S0021904516000356&_rdoc=1&_issn=00219045&md5=4d537a5608534c6fa783e7eea24fc987">dth="134" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021904516000356-si10.gif">dden">de"> and achieve optimality for Smolyak’s algorithm applied to the classical trigonometric interpolation. This can be applied to investigate the sampling numbers for the embedding d="mmlsi11" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516000356&_mathId=si11.gif&_user=111111111&_pii=S0021904516000356&_rdoc=1&_issn=00219045&md5=bd6baf6f57e9621b5fa9c947e67b244c">dth="123" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0021904516000356-si11.gif">dden">de"> for d="mmlsi12" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516000356&_mathId=si12.gif&_user=111111111&_pii=S0021904516000356&_rdoc=1&_issn=00219045&md5=8d3ba71a7037e5d9c726d6f74c417bda" title="Click to view the MathML source">2<q≤∞dden">de">$2<q\le \infty$ where again Smolyak’s algorithm yields the optimal order. The precise decay rates for the sampling numbers in the mentioned situations always coincide with those for the approximation numbers, except probably in the limiting situation d="mmlsi13" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516000356&_mathId=si13.gif&_user=111111111&_pii=S0021904516000356&_rdoc=1&_issn=00219045&md5=4baba807745047fff13cafd26647e9ce" title="Click to view the MathML source">β=γdden">de">$\beta =\gamma$ (including the embedding into d="mmlsi14" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0021904516000356&_mathId=si14.gif&_user=111111111&_pii=S0021904516000356&_rdoc=1&_issn=00219045&md5=861b4f4b77da090c0b70c981c040aca8" title="Click to view the MathML source">L₂(T^d)dden">de">).