Uniform Weak Tractability of Weighted Integration

Abstract

We study a relatively new notion of tractability called “uniform weak tractability” that was recently introduced in (Siedlecki, J. Complex. 29:438–453, 2013 [5]). This notion holds for a multivariable problem iff the information complexity $n(\varepsilon , d)$ of its d-variate component to be solved to within $\varepsilon$ is not an exponential function of any positive power of $\varepsilon ^{-1}$ and/or d. We are interested in necessary and sufficient conditions on uniform weak tractability for weighted integration. Weights are used to control the “role” or “importance” of successive variables and groups of variables. We consider here product weights. We present necessary and sufficient conditions on product weights for uniform weak tractability for two Sobolev spaces of functions defined over the whole Euclidean space with arbitrary smoothness, and of functions defined over the unit cube with smoothness 1. We also briefly consider (s, t)-weak tractability introduced in (Siedlecki and Weimar, J. Approx. Theory 200:227–258, 2015 [6]), and show that as long as $t>1$ then this notion holds for weighted integration defined over quite general tensor product Hilbert spaces with arbitrary bounded product weights.