Liberating the dimension for L2L2-approximation

We consider approximation of ∞∞-variate functions with the error measured in a weighted L2L2-norm. The functions being approximated belong to weighted tensor product spaces with arbitrary weights γuγu. We provide complexity upper bounds and sufficient conditions for polynomial and weak tractabilities expressed in terms of the properties of the weights γuγu and the complexity of the corresponding univariate approximation problem. These tractability conditions are also necessary for important types of weights including product weights.