SIAM Journal on Scientific Computing
Interpolation of functions from Besov-type spaces on Gau&bgr;-Chebyshev grids
Journal of Complexity
Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces
Journal of Approximation Theory
Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials
Journal of Scientific Computing
Generalized Jacobi polynomials/functions and their applications
Applied Numerical Mathematics
A Legendre Petrov-Galerkin method for fourth-order differential equations
Computers & Mathematics with Applications
Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems
SIAM Journal on Scientific Computing
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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.