<p>Change of variable plays an important role in constructing sparse grids for multivariate numerical integration of functions on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$[0,1]^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>. This is a modification of cubature formulae for functions supported in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$[0,1]^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> which yields the same order of convergence. In this paper we prove the necessary and sufficient conditions for the continuity of the change of variable operator in the Sobolev space with mixed derivatives <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\boldsymbol{W}_p^m (\mathbb{R}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">W</mi> </mrow> <mi>p</mi> <mi>m</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1\leq p<\infty$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> . The result is then extended to the spaces on the unit cube <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\boldsymbol{W}_p^m([0,1]^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mrow> <mi mathvariant="bold-italic">W</mi> </mrow> <mi>p</mi> <mi>m</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

Optimality conditions for change of variable operators in Sobolev spaces with mixed derivatives

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发表日期：

2026年2月23日

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V. K. Nguyen

摘要

Change of variable plays an important role in constructing sparse grids for multivariate numerical integration of functions on $[0,1]^d$ ${[0, 1]}^{d}$ . This is a modification of cubature formulae for functions supported in $[0,1]^d$ ${[0, 1]}^{d}$ which yields the same order of convergence. In this paper we prove the necessary and sufficient conditions for the continuity of the change of variable operator in the Sobolev space with mixed derivatives $\boldsymbol{W}_p^m (\mathbb{R}^d)$ $W_{p}^{m} (R^{d})$ with $1\leq p<\infty$ $1 \leq p < \infty$ . The result is then extended to the spaces on the unit cube $\boldsymbol{W}_p^m([0,1]^d)$ $W_{p}^{m} ({[0, 1]}^{d})$ .