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There are universal constants $$C,c>0$$<\/jats:tex-math>\n \n C<\/mml:mi>\n ,<\/mml:mo>\n c<\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (which are specified in the paper) such that $$\\begin{aligned} g^2_n \\le \\frac{C\\log (n)}{n}\\sum \\limits _{k\\ge \\lfloor cn \\rfloor } \\sigma _k^2,\\quad n\\ge 2, \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n g<\/mml:mi>\n n<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msubsup>\n \u2264<\/mml:mo>\n \n \n C<\/mml:mi>\n log<\/mml:mo>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:mfrac>\n \n \u2211<\/mml:mo>\n \n k<\/mml:mi>\n \u2265<\/mml:mo>\n \u230a<\/mml:mo>\n c<\/mml:mi>\n n<\/mml:mi>\n \u230b<\/mml:mo>\n <\/mml:mrow>\n <\/mml:munder>\n \n \u03c3<\/mml:mi>\n k<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msubsup>\n ,<\/mml:mo>\n \n n<\/mml:mi>\n \u2265<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>where $$(\\sigma _k)_{k\\in \\mathbb {N}}$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n \n \u03c3<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n \n k<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the sequence of singular numbers (approximation numbers) of the Hilbert\u2013Schmidt embedding $$\\mathrm {Id}:H(K) \\rightarrow L_2(D,\\varrho _D)$$<\/jats:tex-math>\n \n Id<\/mml:mi>\n :<\/mml:mo>\n H<\/mml:mi>\n \n (<\/mml:mo>\n K<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \u2192<\/mml:mo>\n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n D<\/mml:mi>\n ,<\/mml:mo>\n \n \u03f1<\/mml:mi>\n D<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The algorithm which realizes the bound is a least squares algorithm based on a specific set of sampling nodes. These are constructed out of a random draw in combination with a down-sampling procedure coming from the celebrated proof of Weaver\u2019s conjecture, which was shown to be equivalent to the Kadison\u2013Singer problem. Our result is non-constructive since we only show the existence of a linear sampling operator realizing the above bound. The general result can for instance be applied to the well-known situation of $$H^s_{\\text {mix}}(\\mathbb {T}^d)$$<\/jats:tex-math>\n \n \n H<\/mml:mi>\n mix<\/mml:mtext>\n s<\/mml:mi>\n <\/mml:msubsup>\n \n (<\/mml:mo>\n \n \n T<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in $$L_2(\\mathbb {T}^d)$$<\/jats:tex-math>\n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n (<\/mml:mo>\n \n \n T<\/mml:mi>\n <\/mml:mrow>\n d<\/mml:mi>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with $$s>1\/2$$<\/jats:tex-math>\n \n s<\/mml:mi>\n ><\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We obtain the asymptotic bound $$\\begin{aligned} g_n \\le C_{s,d}n^{-s}\\log (n)^{(d-1)s+1\/2}, \\end{aligned}$$<\/jats:tex-math>\n \n \n \n \n \n \n g<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \u2264<\/mml:mo>\n \n C<\/mml:mi>\n \n s<\/mml:mi>\n ,<\/mml:mo>\n d<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n n<\/mml:mi>\n \n -<\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n log<\/mml:mo>\n \n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n (<\/mml:mo>\n d<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n s<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mtd>\n <\/mml:mtr>\n <\/mml:mtable>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:disp-formula>which improves on very recent results by shortening the gap between upper and lower bound to $$\\sqrt{\\log (n)}$$<\/jats:tex-math>\n \n \n log<\/mml:mo>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msqrt>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. The result implies that for dimensions $$d>2$$<\/jats:tex-math>\n \n d<\/mml:mi>\n ><\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> any sparse grid sampling recovery method does not perform asymptotically optimal.<\/jats:p>","DOI":"10.1007\/s10208-021-09504-0","type":"journal-article","created":{"date-parts":[[2021,4,26]],"date-time":"2021-04-26T19:07:01Z","timestamp":1619464021000},"page":"445-468","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":43,"title":["A New Upper Bound for Sampling Numbers"],"prefix":"10.1007","volume":"22","author":[{"given":"Nicolas","family":"Nagel","sequence":"first","affiliation":[]},{"given":"Martin","family":"Sch\u00e4fer","sequence":"additional","affiliation":[]},{"given":"Tino","family":"Ullrich","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,4,26]]},"reference":[{"key":"9504_CR1","doi-asserted-by":"crossref","unstructured":"A.\u00a0Berlinet and C.\u00a0Thomas-Agnan. 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