{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T15:46:09Z","timestamp":1753890369471,"version":"3.41.2"},"reference-count":35,"publisher":"Frontiers Media SA","license":[{"start":{"date-parts":[[2023,8,24]],"date-time":"2023-08-24T00:00:00Z","timestamp":1692835200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["frontiersin.org"],"crossmark-restriction":true},"short-container-title":["Front. Appl. Math. Stat."],"abstract":"In this article, we study the sampling recovery problem for certain relevant multivariate function classes on the cube [0, 1]d<\/jats:italic><\/jats:sup>, which are not compactly embedded into L<\/mml:mi><\/mml:mrow>\u221e<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>[<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn><\/mml:mrow>]<\/mml:mo><\/mml:mrow><\/mml:mrow>d<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:inline-formula>. Recent tools relating the sampling widths to the Kolmogorov or best m<\/jats:italic>-term trigonometric widths in the uniform norm are therefore not applicable. In a sense, we continue the research on the small smoothness problem by considering limiting smoothness in the context of Besov and Triebel-Lizorkin spaces with dominating mixed regularity such that the sampling recovery problem is still relevant. There is not much information available on the recovery of such functions except for a previous result by Oswald in the univariate case and Dinh D\u0169ng in the multivariate case. As a first step, we prove the uniform boundedness of the \u2113p<\/jats:italic><\/jats:sub>-norm of the Faber coefficients at a fixed level by Fourier analytic means. Using this, we can control the error made by a (Smolyak) truncated Faber series in L<\/mml:mi><\/mml:mrow>q<\/mml:mi><\/mml:mrow><\/mml:msub>(<\/mml:mo>[<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn><\/mml:mrow>]<\/mml:mo><\/mml:mrow><\/mml:mrow>d<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:inline-formula> with q<\/jats:italic> &lt;\u221e. It turns out that the main rate of convergence is sharp. Thus, we obtain results also for S<\/mml:mi><\/mml:mrow>1<\/mml:mn>,<\/mml:mo>\u221e<\/mml:mi><\/mml:mrow>1<\/mml:mn><\/mml:mrow><\/mml:msubsup>F<\/mml:mi>(<\/mml:mo>[<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn><\/mml:mrow>]<\/mml:mo><\/mml:mrow><\/mml:mrow>d<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:inline-formula>, a space \u201cclose\u201d to S<\/mml:mi><\/mml:mrow>1<\/mml:mn><\/mml:mrow>1<\/mml:mn><\/mml:mrow><\/mml:msubsup>W<\/mml:mi>(<\/mml:mo>[<\/mml:mo>0<\/mml:mn>,<\/mml:mo>1<\/mml:mn><\/mml:mrow>]<\/mml:mo><\/mml:mrow><\/mml:mrow>d<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:inline-formula>, which is important in numerical analysis, especially numerical integration, but has rather poor Fourier analytical properties.<\/jats:p>","DOI":"10.3389\/fams.2023.1216331","type":"journal-article","created":{"date-parts":[[2023,8,25]],"date-time":"2023-08-25T07:51:40Z","timestamp":1692949900000},"update-policy":"https:\/\/doi.org\/10.3389\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Lp-Sampling recovery for non-compact subclasses of L\u221e"],"prefix":"10.3389","volume":"9","author":[{"given":"Glenn","family":"Byrenheid","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Serhii","family":"Stasyuk","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tino","family":"Ullrich","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1965","published-online":{"date-parts":[[2023,8,24]]},"reference":[{"journal-title":"Sparse representation of multivariate functions based on discrete point evaluations","year":"2018","author":"Byrenheid","key":"B1"},{"key":"B2","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/s00211-015-0765-y","article-title":"Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions","volume":"134","author":"Hinrichs","year":"2016","journal-title":"Numer Math"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-023-09647-z","article-title":"Path regularity of Brownian motion and Brownian sheet","author":"Kempka","year":"2022","journal-title":"Constr Approx"},{"key":"B4","doi-asserted-by":"crossref","DOI":"10.4171\/085","volume-title":"Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration","author":"Triebel","year":"2010"},{"key":"B5","doi-asserted-by":"crossref","DOI":"10.4171\/107","volume-title":"Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration","author":"Triebel","year":"2012"},{"key":"B6","article-title":"Hyperbolic Cross Approximation","volume-title":"Advanced Courses in Mathematics. 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