Potential theoretic approach to design of accurate formulas for function approximation in symmetric weighted Hardy spaces
Potential theoretic approach to design of accurate formulas for function approximation in symmetric weighted Hardy spaces
复制标题
对称加权 Hardy 空间中函数逼近精确公式设计的潜在理论方法
DOI：
10.1093/imanum/drw022
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发表时间：
2017
影响因子：
2.1
通讯作者：
Tomoaki Okayama and Masaaki Sugihara
中科院分区：
文献类型：
--
作者：
Ken'ichiro Tanaka;Tomoaki Okayama and Masaaki Sugihara
本文提出了一种在加权哈代空间中设计真实的轴上精确插值公式的方法。特别是，我们考虑的哈代空间的功能，是解析的带区域周围的真实的轴，其特征在于由一个权重函数，确定其元素的衰减率在无穷大的邻域。这样的空间被认为是一组函数，这些函数通过在无穷远处实现一定衰减率的变量变换来变换。这种变换的流行的例子是SE-Sinc和DE-Sinc公式的单指数（SE）和双指数（DE）变换，由于在分别具有SE和DE权的加权哈代空间中sinc插值的精确性，它们是非常精确的。然而，不能保证sinc公式在加权哈代空间中是最优的，尽管Sugihara已经证明了它们接近最优。最佳逼近公式的一个显式形式仅在具有某种SE权的加权哈代空间中给出。在一般情况下，迄今为止还没有给出最优公式的显式形式。在加权哈代空间中，采用位势理论方法，得到了一般权函数情形下的几乎最优公式.我们制定的问题，设计一个最佳的公式在每个空间作为一个最优化问题写在一个绿色潜力与外部领域。通过数值求解该优化问题，得到了在每个空间中的一个几乎最优公式。最后，通过数值算例验证了该方法的有效性.特别是，对于DE权重的情况下，由我们的方法设计的公式优于DE-Sinc公式。
We propose a method for designing accurate interpolation formulas on the real axis for the purpose of function approximation in weighted Hardy spaces. In particular, we consider the Hardy space of functions that are analytic in a strip region around the real axis, being characterized by a weight functionthat determines the decay rate of its elements in the neighborhood of infinity. Such a space is considered as a set of functions that are transformed by variable transformations that realize a certain decay rate at infinity. Popular examples of such transformations are given by the single exponential (SE) and double exponential (DE) transformations for the SE-Sinc and DE-Sinc formulas, which are very accurate owing to the accuracy of sinc interpolation in the weighted Hardy spaces with SE and DE weights, respectively. However, it is not guaranteed that the sinc formulas are optimal in weighted Hardy spaces, although Sugihara has demonstrated that they are near optimal. An explicit form for an optimal approximation formula has only been given in weighted Hardy spaces with SE weights of a certain type. In general cases, explicit forms for optimal formulas have not been provided so far. We adopt a potential theoretic approach to obtain almost optimal formulas in weighted Hardy spaces in the case of general weight functions. We formulate the problem of designing an optimal formula in each space as an optimization problem written in terms of a Green potential with an external field. By solving the optimization problem numerically, we obtain an almost optimal formula in each space. Furthermore, some numerical results demonstrate the validity of this method. In particular, for the case of a DE weight, the formula designed by our method outperforms the DE-Sinc formula.