<p>In this paper, we study Hilbert numbers of embedding of Sobolev space of mixed smoothness <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^{s,r}_{\textrm{mix}}({{\mathbb {T}}}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>H</mi> <mtext>mix</mtext> <mrow> <mi>s</mi> <mo>,</mo> <mi>r</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> on the torus <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${{\mathbb {T}}}^d$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L_\infty ({{\mathbb {T}}}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mi>∞</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the Wiener class <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {A}({{\mathbb {T}}}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We obtain the exact asymptotic order of Hilbert numbers of these embeddings and the asymptotic constant for the embedding into <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {A}({{\mathbb {T}}}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We also obtain the asymptotic constant of Hilbert numbers of embedding of Gaussian weighted Sobobev space <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$H^s({{\mathbb {R}}}^d,\gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mi>s</mi> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> into <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {A}({{\mathbb {R}}}^d,\gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo>,</mo> <mi>γ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> which is a counterpart of the Wiener class <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {A}({{\mathbb {T}}}^d)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Hilbert numbers of Sobolev spaces with mixed smoothness in the sup-norm

期刊论文
发表日期：

2025年11月6日

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Van Kien Nguyen

摘要

In this paper, we study Hilbert numbers of embedding of Sobolev space of mixed smoothness $H^{s,r}_{\textrm{mix}}({{\mathbb {T}}}^d)$ $H_{mix}^{s, r} (T^{d})$ on the torus ${{\mathbb {T}}}^d$ $T^{d}$ into $L_\infty ({{\mathbb {T}}}^d)$ $L_{\infty} (T^{d})$ and the Wiener class $\mathcal {A}({{\mathbb {T}}}^d)$ $A (T^{d})$ . We obtain the exact asymptotic order of Hilbert numbers of these embeddings and the asymptotic constant for the embedding into $\mathcal {A}({{\mathbb {T}}}^d)$ $A (T^{d})$ . We also obtain the asymptotic constant of Hilbert numbers of embedding of Gaussian weighted Sobobev space $H^s({{\mathbb {R}}}^d,\gamma )$ $H^{s} (R^{d}, γ)$ into $\mathcal {A}({{\mathbb {R}}}^d,\gamma )$ $A (R^{d}, γ)$ which is a counterpart of the Wiener class $\mathcal {A}({{\mathbb {T}}}^d)$ $A (T^{d})$ .