Equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in Lp

We consider $\gamma$-weighted anchored and ANOVA spaces of functions with mixed first order partial derivatives bounded in a weighted $L_p$ norm with $1 \leq p \leq \infty$. The domain of the functions is $D^d$, where $D \subseteq \mathbb{R}$ is a bounded or unbounded interval. We provide conditions on the weights $\gamma$ that guarantee that anchored and ANOVA spaces are equal (as sets of functions) and have equivalent norms with equivalence constants uniformly or polynomially bounded in $d$. Moreover, we discuss applications of these results to integration and approximation of functions on $D^d$.

[1]  Friedrich Pillichshammer,et al.  Bounds for the weighted Lp discrepancy and tractability of integration , 2003, J. Complex..

[2]  Klaus Ritter,et al.  On embeddings of weighted tensor product Hilbert spaces , 2015, J. Complex..

[3]  Grzegorz W. Wasilkowski,et al.  On tractability of approximation in special function spaces , 2013, J. Complex..

[4]  Henryk Wozniakowski,et al.  On decompositions of multivariate functions , 2009, Math. Comput..

[5]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[6]  Aicke Hinrichs,et al.  Tractability properties of the weighted star discrepancy , 2008, J. Complex..

[7]  Grzegorz W. Wasilkowski.  Tractability of approximation of ∞-variate functions with bounded mixed partial derivatives , 2014, J. Complex..

[8]  Michael Gnewuch,et al.  Embeddings of weighted Hilbert spaces and applications to multivariate and infinite-dimensional integration , 2017, J. Approx. Theory.

[9]  Frances Y. Kuo,et al.  Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients , 2012, 1208.6349.

[10]  Henryk Wozniakowski,et al.  When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? , 1998, J. Complex..

[11]  E. Novak,et al.  Tractability of Multivariate Problems , 2008 .

[12]  Henryk Wozniakowski,et al.  Good Lattice Rules in Weighted Korobov Spaces with General Weights , 2006, Numerische Mathematik.

[13]  Aicke Hinrichs,et al.  Equivalence of anchored and ANOVA spaces via interpolation , 2016, J. Complex..

[14]  Christoph Aistleitner,et al.  Tractability results for the weighted star-discrepancy , 2013, J. Complex..

[15]  Grzegorz W. Wasilkowski,et al.  A note on equivalence of anchored and ANOVA spaces; lower bounds , 2017, J. Complex..

[16]  Grzegorz W. Wasilkowski,et al.  On equivalence of weighted anchored and ANOVA spaces of functions with mixed smoothness of order one in L1 or L∞ , 2016, J. Complex..