Optimal Order of Convergence and (In)Tractability of Multivariate Approximation of Smooth Functions

We study the approximation problem for C ∞ functions f :[0,1] d →ℝ with respect to a W p m -norm. Here, m =[ m , m ,…, m ], d times, with the norm of the target space defined in terms of up to m partial derivatives with respect to all d variables. The optimal order of convergence is infinite, hence...

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Vydáno v:Constructive approximation Ročník 30; číslo 3; s. 457 - 473
Hlavní autoři: Novak, Erich, Woźniakowski, Henryk
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer-Verlag 01.12.2009
Témata:
Analysis
Mathematics
Mathematics and Statistics
Numerical Analysis
Tractability
Curse of dimensionality
41A25
41A63
Approximation of smooth functions
65Y20
Rate of convergence
ISSN:0176-4276, 1432-0940
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Popis
Shrnutí:We study the approximation problem for C ∞ functions f :[0,1] d →ℝ with respect to a W p m -norm. Here, m =[ m , m ,…, m ], d times, with the norm of the target space defined in terms of up to m partial derivatives with respect to all d variables. The optimal order of convergence is infinite, hence excellent, but the problem is still intractable and suffers from the curse of dimensionality if m ≥1. This means that the order of convergence supplies incomplete information concerning the computational difficulty of a problem. For m =0 and p =2, we prove that the problem is not polynomially tractable, but that it is weakly tractable.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-009-9069-8