Polynomial series versus sinc expansions for functions with corner or endpoint singularities
Polynomial series versus sinc expansions for functions with corner or endpoint singularities
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DOI：
10.1016/0021-9991(86)90031-8
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发表时间：
1986-05
影响因子：
4.1
通讯作者：
J. Boyd
中科院分区：
文献类型：
--
作者：
J. Boyd
多项式与 SING 系列 267 n- 是多项式的最佳选择，并促使人们寻找替代方案。成功的选项使用了一系列 x 的超越函数项。 Stenger [l] 表明可以通过三步过程创建近似值，其误差以指数方式而不是代数方式减小。第一步是通过映射 x= tanh (ky)(3) 将区间 x E [-1, 11 转换为 y E [-co, cc]，其中 k 是任意比例因子。第二步，选择网格间距h，然后通过“正弦展开”或“惠特克基数”逼近来近似f(y[x])，即f(y)=ff(3)sine(CY-31/h)； YE [--co, 001,=--co 其中“sin”函数由 sine (z) 5 sin (7rz)/(rcz) 定义。(5)最后一步是截断 (4) 中的无限级数，以便我们对有限数量的网格点 N 求和。Stenger [l] 继续表明正弦展开式可用于求解微分方程和积分方程，但为了简单起见，我们假设 f (x [y]) 是已知函数。
POLYNOMIALS VERSUS SING SERIES 267 n-is the best possible for polynomials, and has prompted a search for alternatives. The successful options use a series of terms which are transcendental functions of x. Stenger [l] showed that it is possible to create an approximation through a three-step procedure whose error decreases exponentially rather than algebraically. The first step is to transform the interval x E [-1, 11 to y E [-co, cc] through the mapping x= tanh (ky)(3) where k is an arbitrary scale factor. The second step is to choose a grid spacing h and then approximate f (y [x]) through the “sine expansion” or “Whittaker cardinal” approximation, which is f (y)= ff (3) sine (CY-31/h); YE [--co, 001,=--co where the “sin? function is defined by sine (z) 5 sin (7rz)/(rcz).(5)The final step is to truncate the infinite series in (4) so that we sum over a finite number of grid points N. Stenger [l] goes on to show that the sine expansion can be used to solve differential and integral equations, but for simplicity, we will assume f (x [y]) is a known function.