Quasi-interpolation for linear functional data
Quasi-interpolation for linear functional data
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DOI：
10.1016/j.cam.2012.02.028
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发表时间：
2012-07
期刊：
影响因子：
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通讯作者：
Wenwu Gao;Zongmin Wu
中科院分区：
文献类型：
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作者：
Wenwu Gao;Zongmin Wu
拟插值在文献中得到了广泛的研究。然而，大多数拟插值的研究通常只针对离散函数值（或离散函数值的有限线性组合）。请注意，在实际应用中，更常见的是，我们可以对线性函数数据（某些微分方程右侧的离散值）而不是离散函数值（例如，遥感、地震数据等）。因此，研究线性函数型数据的拟插值问题更有意义。本文的主要结果是提出这样一个准插值方案。文中还给出了格式的误差估计。基于误差估计，人们可以找到一个准插值，提供了一个最佳的逼近阶相对于光滑的右手边的微分方程。该格式可以应用于微分方程的数值求解、构造李雅普诺夫函数等多种场合，文末给出了相应的例子。
Quasi-interpolation has been studied extensively in the literature. However, most studies of quasi-interpolation are usually only for discrete function values (or a finite linear combination of discrete function values). Note that in practical applications, more commonly, we can sample the linear functional data (the discrete values of the right-hand side of some differential equations) rather than the discrete function values (e.g., remote sensing, seismic data, etc). Therefore, it is more meaningful to study quasi-interpolation for the linear functional data. The main result of this paper is to propose such a quasi-interpolation scheme. Error estimate of the scheme is also given in the paper. Based on the error estimate, one can find a quasi-interpolant that provides an optimal approximation order with respect to the smoothness of the right-hand side of the differential equation. The scheme can be applied in many situations such as the numerical solution of the differential equation, construction of the Lyapunov function and so on. Respective examples are presented in the end of this paper.