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首页> 外文期刊>SIAM Journal on Numerical Analysis >UNIFORM APPROXIMATION OF PIECEWISE r-SMOOTH AND GLOBALLY CONTINUOUS FUNCTIONS
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UNIFORM APPROXIMATION OF PIECEWISE r-SMOOTH AND GLOBALLY CONTINUOUS FUNCTIONS

机译：分段r-光滑和全局连续函数的统一逼近

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摘要

We study the uniform (Chebyshev) approximation of continuous and piecewise rsmooth (r ≥ 2) functions f : [0, T] → R with a finite number of singular points. The approximation algorithms use only n function values at adaptively or nonadaptively chosen points. We construct a nonadaptive algorithm Anon r,n that, for the functions with at most one singular point, enjoys the best possible convergence rate nr. This is in sharp contrast to results concerning discontinuous functions. For r ≥ 3, this optimal rate of convergence holds only in the asymptotic sense, i.e., it occurs only for sufficiently large n that depends on f in a way that is practically impossible to verify. However, it is enough to modify Anon r,n by using (r +1)(r 1)/2extra function evaluations to obtain an adaptive algorithm Aada r,n with error satisfying f Aada r,n fC ≤ CrTrf(r)∞L nr for all n ≥ n0 and n0 independent of f. This result cannot be achieved for functions with mre than just one singular point.However, the convergence rate nr can be recovered asymptotically by a nonadaptive algorithm A non r,n that is a slightly modified Anon r,n . Specifically, lim supn→∞ f A non r,n f C · nr ≤ CrTrf(r)∞L for all r-smooth functions f with finitely many singular points.
机译：我们研究具有有限个奇点的连续和分段rsmooth（r≥2）函数f：[0，T]→R的一致（Chebyshev）逼近。近似算法仅在自适应或非自适应选择的点上使用n个函数值。我们构造了一种非自适应算法Anon r，n，对于具有最多一个奇点的函数，其具有最佳的收敛速度nr。这与有关不连续功能的结果形成鲜明对比。当r≥3时，该最佳收敛速度仅在渐近意义上成立，即，仅当依赖于f的足够大的n出现时，这种收敛速度实际上是无法验证的。但是，通过使用（r +1）（r 1）/ 2附加函数求值来修改Anon r，n足以获得误差满足​​f Aada r，n fC≤CrTrf（r）∞的自适应算法Aada r，n所有n≥n0和n0的L nr独立于f。对于仅具有一个奇点的mre函数，无法获得此结果。但是，可以通过非自适应算法A non r，n（稍微修改了Anon r，n）渐近地恢复收敛速度nr。具体来说，对于所有具有有限多个奇点的r光滑函数f，lim supn→∞f A non r，n f C·nr≤CrTrf（r）∞L。

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