Embeddings for Infinite-Dimensional Integration and $L_2$-Approximation with Increasing Smoothness

Abstract

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.

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Cited by in corpus (5)

• Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration
• Infinite-dimensional integration and $L^2$-approximation on Hermite spaces
• How anisotropic mixed smoothness affects the decay of singular numbers of Sobolev embeddings
• Infinite-Variate $L^2$-Approximation with Nested Subspace Sampling
• Equivalence between Sobolev spaces of first-order dominating mixed smoothness and unanchored ANOVA spaces on ${\mathbb{R}}^d$