On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in L1 or L∞

Abstract

a b s t r a c tWe consider weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in L 1 or L ∞ norms. We provide conditions on the weights under which the corresponding spaces have equivalent norms with constants independent of, or only polynomially dependent on the number of variables.

Truncation dimension for linear problems on multivariate function spaces

“…As in [3], one can show that the spaces F s,p 1 ,p 2 ,γ and F s,p 1 ,p 2 ,γ,ω as sets of functions are equal if and only if…”

“…In this section, we begin by recalling the definitions and basic properties of weighted anchored Sobolev spaces of s-variate functions with mixed partial derivatives of order one bounded in L p -norm. More detailed information can be found in [3,4,14]. Such spaces have often been assumed in the context of quasi-Monte Carlo methods.…”

Truncation Dimension for Function Approximation

“…In particular we have ε 10 −1 10 −2 10 −3 10 −4 10 −5 10 −6 dim trnc (ε) 3 Consider next p > 1. Unlike in the case p = 1, we do not know the exact values of the truncation dimension.…”

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

“…Remark 4.5. The algorithms used to derive the error bound (21) belong to the class of interlaced scrambled polynomial lattice rules. For a given prime base b an interlaced scrambled polynomial lattice rule Q of order r consisting of b m points in dimension s is constructed in the following way: First an ordinary polynomial lattice rule with b m points in dimension rs is generated and afterwards the points are randomized via Owen's b-ary digit scrambling [39].…”