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• W4386167136 abstract "In this article, we study the sampling recovery problem for certain relevant multivariate function classes on the cube [0, 1] d , which are not compactly embedded into <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M1><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>∞</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:math> . Recent tools relating the sampling widths to the Kolmogorov or best m -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 Dũng in the multivariate case. As a first step, we prove the uniform boundedness of the ℓ p -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 <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M2><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:math> with q &lt;∞. It turns out that the main rate of convergence is sharp. Thus, we obtain results also for <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M3><mml:msubsup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>∞</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mi>F</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:math> , a space “close” to <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML id=M4><mml:msubsup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mi>W</mml:mi><mml:mrow><mml:mo stretchy=false>(</mml:mo><mml:mrow><mml:msup><mml:mrow><mml:mrow><mml:mo>[</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>]</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy=false>)</mml:mo></mml:mrow></mml:math> , which is important in numerical analysis, especially numerical integration, but has rather poor Fourier analytical properties." @default.
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• W4386167136 date "2023-08-24" @default.
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• W4386167136 title "Lp-Sampling recovery for non-compact subclasses of L∞" @default.
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