We present new cubature formulas for multivariate periodic functions with respect to each variable with period 1 and some smoothness α > max(1/q, 1/2), where 1 ≤ q ≤ ∞. The new cubature formulas are exact for polynomials with frequencies from the corresponding hyperbolic cross, and the upper bound of the worst-case error is <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7774_Article_IEq1.gif" Format="GIF" Height="33" Rendition="HTML" Resolution="72" Type="Linedraw" Width="194" /> </InlineMediaObject> <EquationSource Format="TEX">$O\left({N}^{-\alpha }{\left(\text{log}N\right)}^{\left(\alpha +1/\widetilde{q}\right)\left(n-1\right)}\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>O</mi> <mfenced close=")" open="("> <msup> <mrow> <mi>N</mi> </mrow> <mrow> <mo>-</mo> <mi>α</mi> </mrow> </msup> <msup> <mrow> <mfenced close=")" open="("> <mtext>log</mtext> <mi>N</mi> </mfenced> </mrow> <mrow> <mfenced close=")" open="("> <mi>α</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mover accent="true"> <mi>q</mi> <mo stretchy="false">~</mo> </mover> </mfenced> <mfenced close=")" open="("> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mfenced> </mrow> </msup> </mfenced> </mrow> </math></EquationSource> </InlineEquation> for periodic functions in the Sobolev spaces <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7774_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">${W}_{q}^{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>W</mi> <mrow> <mi>q</mi> </mrow> <mi>α</mi> </msubsup> </math></EquationSource> </InlineEquation>, where N is the number of nodes, <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10958_2025_7774_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">$\widetilde{q}=\text{min}\left(q,2\right)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>q</mi> <mo stretchy="false">~</mo> </mover> <mo>=</mo> <mtext>min</mtext> <mfenced close=")" open="("> <mi>q</mi> <mo>,</mo> <mn>2</mn> </mfenced> </mrow> </math></EquationSource> </InlineEquation>, n is the dimension. The feature of our methods is that the error is represented in terms of Fourier coefficients.

New Cubature Formulas for Sobolev Spaces with Dominant Mixed Derivative

期刊论文
发表日期：

2025年6月7日

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Serzhan Bassarov,
Erlan Nursultanov

摘要

We present new cubature formulas for multivariate periodic functions with respect to each variable with period 1 and some smoothness α > max(1/q, 1/2), where 1 ≤ q ≤ ∞. The new cubature formulas are exact for polynomials with frequencies from the corresponding hyperbolic cross, and the upper bound of the worst-case error is $O\left({N}^{-\alpha }{\left(\text{log}N\right)}^{\left(\alpha +1/\widetilde{q}\right)\left(n-1\right)}\right)$ $O (N^{- α}, {(log, N)}^{(α + 1 / \tilde{q}) (n - 1)})$ for periodic functions in the Sobolev spaces ${W}_{q}^{\alpha }$ $W_{q}^{α}$ , where N is the number of nodes, $\widetilde{q}=\text{min}\left(q,2\right)$ $\tilde{q} = min (q, 2)$ , n is the dimension. The feature of our methods is that the error is represented in terms of Fourier coefficients.