论文标题
在奇异和振荡函数的近似中，高音插入是否有效？
Is hyperinterpolation efficient in the approximation of singular and oscillatory functions?
论文作者
论文摘要
在众多应用中，单数和振荡功能的特征。此类功能的高临界性近似应极大地帮助我们开发用于解决应用数学问题的高级方法。本文表明，通过使用某些数值集成方法评估$ l^2 $正交投影系数获得的系数的离散投影方法，高插值可能是近似单数和振动函数的效率低下。对于令人满意的精度，必须使用相对较大的数值集成点。此外，本着产品整合的精神，我们提出了对这种近似值的高度交流的有效修改。所提出的近似方案（称为有效的超插值）可达到令人满意的精度，而数值集成点比原始方案更少。新近似方案的实现相对容易。还提供了定理来解释在这种近似值中与原始方案相比的高效超闭合的表现，其函数分别属于$ l^1（ω）$，$ l^2（ω）$和$ \ MATHCAL {C}（ω）（ω）$ Space。这些定理以及间隔和球体上的数值实验表明，当数值积分点的量受到限制时，在近似值中，有效的高接个性在如此近似中具有更好的精度。
Singular and oscillatory functions feature in numerous applications. The high-accuracy approximation of such functions shall greatly help us develop high-order methods for solving applied mathematics problems. This paper demonstrates that hyperinterpolation, a discrete projection method with coefficients obtained by evaluating the $L^2$ orthogonal projection coefficients using some numerical integration methods, may be inefficient for approximating singular and oscillatory functions. A relatively large amount of numerical integration points are necessary for satisfactory accuracy. Moreover, in the spirit of product-integration, we propose an efficient modification of hyperinterpolation for such approximation. The proposed approximation scheme, called efficient hyperinterpolation, achieves satisfactory accuracy with fewer numerical integration points than the original scheme. The implementation of the new approximation scheme is relatively easy. Theorems are also given to explain the outperformance of efficient hyperinterpolation over the original scheme in such approximation, with the functions assumed to belong to $L^1(Ω)$, $L^2(Ω)$, and $\mathcal{C}(Ω)$ spaces, respectively. These theorems, as well as numerical experiments on the interval and the sphere, show that efficient hyperinterpolation has better accuracy in such approximation than the original one when the amount of numerical integration points is limited.
