We study integration and $L_2$-approximation on countable tensor products of function spaces of increasing smoothness. We obtain upper and lower bounds for the minimal errors, which are sharp in many cases including, e.g., Korobov, Walsh, Haar, and Sobolev spaces. For the proofs we derive embedding theorems between spaces of increasing smoothness and appropriate weighted function spaces of fixed smoothness.

中文翻译：

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用于无穷维集成的嵌入和 ${\mathrm{大号}}_2$-随着平滑度的提高而近似

我们研究整合和 ${\mathrm{大号}}_2$-增加平滑度的函数空间的可数张量积的逼近。我们获得最小误差的上限和下限，在许多情况下，例如Korobov，Walsh，Haar和Sobolev空间，这些误差非常明显。对于证明，我们推导了在增加平滑度的空间和适当的固定平滑度的加权函数空间之间的嵌入定理。