Counting Via Entropy: New Preasymptotics for the Approximation Numbers of Sobolev Embeddings
Counting Via Entropy: New Preasymptotics for the Approximation Numbers of Sobolev Embeddings
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DOI：
10.1137/16m106580x
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发表时间：
2015-05
期刊：
影响因子：
--
通讯作者：
T. Kühn;Sebastian Mayer;T. Ullrich
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文献类型：
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作者：
T. Kühn;Sebastian Mayer;T. Ullrich
在本文中，我们揭示了周期Sobolev型空间的近似数与嵌入的熵数$\textrm{id}: \ell_{\|\cdot\|}^d \to \ell_\infty^d$之间的新联系，其中傅里叶系数的平滑权是由$\mathbb{R}^d$上的（拟）范数$\|\cdot\|$诱导的。这种联系产生了$L_2$和$H^1$中各向同性Sobolev空间、解析函数空间和Gevrey型空间的近似数的预渐近误差界，它们在伽辽金方法的背景下得到了应用。此外，我们还观察到某些Gevrey型空间的近似数与占统治地位的混合光滑空间的近似数在前渐近上几乎相同。这一观察结果可以被利用，例如，电子薛定谔方程的伽辽金方案，其中存在混合正则性。
In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier coefficients are induced by a (quasi-)norm $\|\cdot\|$ on $\mathbb{R}^d$, and entropy numbers of the embedding $\textrm{id}: \ell_{\|\cdot\|}^d \to \ell_\infty^d$. This connection yields preasymptotic error bounds for approximation numbers of isotropic Sobolev spaces, spaces of analytic functions, and spaces of Gevrey type in $L_2$ and $H^1$, which find application in the context of Galerkin methods. Moreover, we observe that approximation numbers of certain Gevrey type spaces behave preasymptotically almost identical to approximation numbers of spaces of dominating mixed smoothness. This observation can be exploited, for instance, for Galerkin schemes for the electronic Schrodinger equation, where mixed regularity is present.