An adaptive algorithm for weighted approximation of singular functions over ?

摘要

We study the ω -weighted L~p approximation (1 ≤ p ≤ ∞) of piecewise r-smooth functions f: ? → ?. Approximations Anf are based on n values of f at points that can be chosen adaptively. Assuming that the weight Σ is Riemann integrable on any compact interval and asymptotically decreasing, a necessary condition for the error of approximation to be of order n~(-r) is that ∥ Σ∥L~(1/γ) < ∞, where γ = r+1/p. For the class Wγ of globally γ-smooth functions, this condition is also sufficient. Indeed, we show a nonadaptive algorithm P~* _n with the worst case error supf(eqution presented) n-rSuch worst case result does not hold in general for the class of piecewise r-smooth functions. However, if p < ∞ and the class is restricted to F?~1 _r of functions with at most one singularity and uniformly bounded singularity jumps, then an adaptive algorithm A ~* _n can be constructed whose worst case error satisfies sup f (eqution presented) A modification of A.n gives an adaptive algorithm A~* _n such that the error (eqution presented) max (eqution) is of order n~(-r) for any function f with finitely many singular points and with no restrictions on the jumps. For those results to hold, the use of adaption and p < ∞ is necessary. Yet similar results can be obtained if the error is measured in the weighted Skorohod metric instead of the weighted L∞ norm.