Uniform Weak Tractability of Multivariate Problems

In this dissertation we introduce a new notion of tractability which is called uniform weak tractability. We give necessary and sufficient conditions on uniform weak tractability of homogeneous linear tensor product problems in the worst case, average case and randomized settings. We then turn to the study of approximation problems defined over spaces of functions with varying regularity with respect to successive variables. In the worst case setting we study approximation problems defined over suitable Korobov and Sobolev spaces. In the average case setting we study approximation problems defined over the space of continuous functions C([0, 1]^d ) equipped with a zero-mean Gaussian measure whose covariance operator is given by Euler or Wiener integrated process. We establish necessary and sufficient conditions on uniform weak tractability of those problems in terms of their regularity parameters.