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Error analysis for frequency-dependent interpolation formulas using first derivatives

机译：使用一阶导数的频率相关插值公式的误差分析

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摘要

We call attention to the entire errors which result when frequency-dependent interpolation formulas are utilized to approximate oscillatory functions f(x) at some x on the domain of interest with a frequency ω of the form,f(x)=f~1(x)cos(ωx)+f_2(x) sin(ωx),where the functions F~1 and f_2 are smooth enough to be approximated by polynomials. The interpolation formulas to be considered utilize not only the pointwise values of the function f but also of its derivative f′ at two or three nodes on a closed and bounded interval. In particular, investigations about the interpolation formulas I (or I) using three equally spaced nodes (or three unequally spaced nodes) enable us to construct I (or I)-related composite formulas which are obtained from applying the formulas I (or I) onto subintervals where the union of all the subintervals is the domain of interest. Numerical results show that newly constructed composite formulas are superior in their accuracy to other approximations to interpolate the oscillatory functions. Finally, the entire errors with respect to the interpolation formulas using the derivative information at two (or three) nodes are obtained.
机译：我们提请注意整个误差，这些误差是当使用频率相关的插值公式来近似感兴趣的域上某些x处的振动函数f（x）时，其频率ω的形式为f（x）= f〜1（ x）cos（ωx）+ f_2（x）sin（ωx），其中函数F〜1和f_2足够平滑，可以通过多项式近似。要考虑的插值公式不仅利用函数f的逐点值，而且还利用封闭和有界区间上两个或三个节点处的函数f'的逐点值。特别地，使用三个等距节点（或三个不等距节点）对插值公式I（或I）的研究使我们能够构造与I（或I）相关的合成公式，这些公式是通过应用公式I（或I）获得的到所有子区间的并集是关注域的子区间。数值结果表明，新构建的复合公式在精度上优于插值振荡函数的其他近似公式。最终，获得关于在两个（或三个）节点上使用导数信息的内插公式的全部误差。

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