Introduction to combinatorial theory
Introduction to combinatorial theory
Information-based complexity
Linear vs. nonlinear algorithms for linear problems
Journal of Complexity
Approximation of infinitely differentiable multivariate functions is intractable
Journal of Complexity
Approximation rates for the hierarchical tensor format in periodic Sobolev spaces
Journal of Complexity
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We investigate optimal linear approximations (approximation numbers) in the context of periodic Sobolev spaces H^s(T^d) of fractional smoothness s0 for various equivalent norms including the classical one. The error is always measured in L"2(T^d). Particular emphasis is given to the dependence of all constants on the dimension d. We capture the exact decay rate in n and the exact decay order of the constants with respect to d, which is in fact polynomial. As a consequence we observe that none of our considered approximation problems suffers from the curse of dimensionality. Surprisingly, the square integrability of all weak derivatives up to order three (classical Sobolev norm) guarantees weak tractability of the associated multivariate approximation problem.