论文标题
关于高交中的正交精确性
On the quadrature exactness in hyperinterpolation
论文作者
论文摘要
本文研究了正交精确性在高中间近似方案中的作用。构建$ n $的高度间隔剂需要一个正重的正交规则，具有精确性$ 2N $。当所需的精确度$ 2N $放松至$ n+k $，$ 0 <k \ leq n $时，我们检查了这种近似值的行为。在Marcinkiewicz-Zygmund不平等的协助下，我们确认，精确性删除的超插值操作员的$ l^2 $规范受$ n $独立的常数范围，并且如果$ n \ rightarrow \ infty $ k $与$ n $ N $ N $ N.因此，可以显着丰富用于构建高音中质体的候选二次规则，并且可以大大减少正交点的数量。作为潜在的成本，这种放松可能会根据降低的正交精确性度降低过度中间的收敛速度。我们的理论结果是通过数值实验对三个最著名的正交规则的数字实验所主张的：高斯正交，clenshaw-curtis quadrature和球形$ t $ designs。
This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree $n$ requires a positive-weight quadrature rule with exactness degree $2n$. We examine the behavior of such approximation when the required exactness degree $2n$ is relaxed to $n+k$ with $0<k\leq n$. Aided by the Marcinkiewicz--Zygmund inequality, we affirm that the $L^2$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of $n$, and this approximation scheme is convergent as $n\rightarrow\infty$ if $k$ is positively correlated to $n$. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness. Our theoretical results are asserted by numerical experiments on three of the best-known quadrature rules: the Gauss quadrature, the Clenshaw--Curtis quadrature, and the spherical $t$-designs.
