Equivalence of anchored and ANOVA spaces via interpolation

Abstract

We consider weighted anchored and ANOVA spaces of functions with first order mixed derivatives bounded in L p . Recently, Hefter, Ritter and Wasilkowski established conditions on the weights in the cases p = 1 and p = ∞ which ensure equivalence of the corresponding norms uniformly in the dimension or only polynomially dependent on the dimension. We extend these results to the whole range of p ∈ [1, ∞]. It is shown how this can be achieved via interpolation.MSC: 65D30,65Y20,41A63,41A55

On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in L1 or L∞Hefter1, Ritter2, Wasilkowski3 2016Journal of Complexity222630View full textAdd to dashboardCite

“…The proof techniques used in this paper are not strong enough to get sharp results for p ∈ (1, ∞). However, the authors of a forthcoming paper [8] will present results for p ∈ (1, ∞) that are obtained by using interpolation and our results for p = 1 and p = ∞.…”

Embeddings for infinite-dimensional integration and L2-approximation with increasing smoothness

“…is a sufficient condition for (21) ν,j∈N α −1 ν,j · ν σ < ∞ to hold. A necessary condition also permits ρ = 1/ ln(a 1 ).…”

Truncation dimension for linear problems on multivariate function spaces

“…Moreover, following the approach in [2], one can provide exact formulas for the norms of the embeddings for p 1 , p 2 ∈ {1, ∞} and next, using interpolation theory (as in [4], see also [2]), derive upper bounds for arbitrary values of p 1 and p 2 .…”

“…It was shown in [4] for product weights and in [6] for a number of different types of weights that the upper bounds in Corollary 9 are sharp. Suppose now that ∞ j=1 γ j < ∞.…”