Random Gel′fand Approximation Between Anisotropic Spaces and Function Spaces With Mixed Smoothness

This paper investigates the complexity associated with function approximation between anisotropic spaces and spaces of mixed smoothness within the framework of adaptive Monte Carlo methods. Specifically, we analyze the information complexity for approximating function classes of mixed smoothness (1 ≤ p < ∞) in anisotropic spaces (1 ≤ q < ∞) through adaptive Monte Carlo techniques, as well as the reverse problem of approximating function classes of anisotropic spaces (1 ≤ p < ∞) in mixed‐smoothness spaces (1 ≤ q < ∞). By employing discretization methods and exploiting pseudo‐s‐scale properties, we further estimate the precise asymptotic orders of the associated approximation problems. The results reveal the intrinsic computational complexity of adaptive Monte Carlo approximation in high‐dimensional settings and provide a rigorous theoretical foundation for the design and analysis of efficient Monte Carlo–type algorithms.