Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration

Abstract

We study embeddings and norm estimates for tensor products of weighted reproducing kernel Hilbert spaces. These results lead to a transfer principle that is directly applicable to tractability studies of multivariate problems as integration and approximation, and to their infinite-dimensional counterparts. In an application we consider weighted tensor product Sobolev spaces of mixed smoothness of any integer order, equipped with the classical, the anchored, or the ANOVA norm. Here we derive new results for multivariate and infinite-dimensional integration.

Explicit error bounds for randomized Smolyak algorithms and an application to infinite-dimensional integration

“…In Theorem 18 we provide a sharp result on randomized infinite-dimensional integration on weighted reproducing kernel Hilbert spaces that parallels the sharp result on deterministic infinite-dimensional integration stated in [24,Theorem 5.1]. Results from [23] and from [52] in combination with Theorem 7 rigorously establish the sharp randomized result in the special case where the weighted reproducing kernel Hilbert space is based on an anchored univariate kernel.…”

“…The next corollary on infinite-dimensional integration on weighted Korobov spaces in the randomized setting parallels [24,Theorem 5.5], which discusses the deterministic setting.…”

“…For a general reproducing kernel k 1 we need to find a suitably associated reproducing kernel k a anchored in a and satisfying γk a = (k a ) γ for all γ > 0 to employ the embedding machinery from [24] to obtain the desired result (52). To this purpose we consider the bounded linear functional ξ : Notice that · 1,a is induced by the symmetric bilinear form f, g 1,a := ξ(f ) · ξ(g).…”

“…The class A res contains not only algorithms of the form (48), but also adaptive and non-linear algorithms. In the proof of Theorem 18 we employ the function space embeddings from [24], which allows us to transfer results for linear algorithms from the case of anchored kernels to the general case. Hence we can conclude that the upper bound for λ r (K γ ∞ ) in (52) holds also if we admit adaptive linear algorithms of the form (48) for infinite-dimensional as well as for univariate integration, but we do not know whether this is still the case if we admit non-linear algorithms.…”

“…The analysis relies on explicit cost bounds for deterministic Smolyak algorithms from [64]. In [25,Theorem 4.5] the result is extended to weighted (not necessarily anchored) reproducing kernel Hilbert spaces (relying on the embedding tools from [24]) and to spaces of increasing smoothness. Now one may use Theorem 7 to establish a corresponding result to [63, Corollary 9] for weighted anchored spaces in the randomized setting and may generalize it to nonanchored weighted spaces and to spaces of increasing smoothness via the embedding results established in [24,25].…”