Uniform weak tractability of multivariate problems with increasing smoothness

We study dd-variate approximation problems with varying regularity with respect to successive variables. The varying regularity is described by a sequence of real numbers {rk}k∈N{rk}k∈N satisfying 0≤r1≤r2≤r3≤⋯.0≤r1≤r2≤r3≤⋯. We mainly consider algorithms that use finitely many continuous linear functionals. 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)C([0,1]d) equipped with a zero-mean Gaussian measure whose covariance operator is given by an Euler or Wiener integrated process. We establish necessary and sufficient conditions on uniform weak tractability of those problems in terms of their regularity parameters {rk}k∈N{rk}k∈N.