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Sparse Spectral Approximations of High-Dimensional Problems Based on Hyperbolic Cross

Hyperbolic cross approximations by some classical orthogonal polynomials/functions in both bounded and unbounded domains are considered in this paper. Optimal error estimates in proper anisotropic weighted Korobov spaces for both regular hyperbolic cross approximations and optimized hyperbolic cross approximations are established. These fundamental approximation results indicate that spectral methods based on hyperbolic cross approximations can be effective for treating certain high-dimensional problems and will serve as basic tools for analyzing sparse spectral methods in high dimensions.