球基函数的L^p Bernstein估计和逼近
L^p Bernstein estimates and approximation by spherical basis functions
摘要：
本文的目的是建立L^p误差估计，Bernstein不等式，和逆定理的逼近空间位于分散的网站上的单位n-球的球面基函数。特别地，伯恩斯坦不等式根据函数本身的最小分离度和L^p范数估计了该空间中函数的L^p贝塞尔-位势Sobolev范数。证明中的一个重要步骤是用系数的l^p范数来度量逼近空间中函数的L^p稳定性。作为Bernstein不等式的一个应用，我们得到了L^P范数下SBF逼近的逆定理.最后，我们给出了n-球面上Besov空间的一个新的特征。
英文摘要：
The purpose of this paper is to establish L^p error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates L^p Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the L^p norm of the function itself. An important step in its proof involves measuring the L^p stability of functions in the approximating space in terms of the l^p norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the L^P norm. Finally, we give a new characterization of Besov spaces on the n-sphere in terms of spaces of SBFs.
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