最优化的勒贝格常数用于球面上的最小二乘多项式逼近
Optimal Lebesgue constants for least squares polynomial approximation on the (hyper)sphere
AI总结：
研究单位球上最小二乘多项式均匀逼近的最优勒贝格常数，探讨高次多项式逼近的均匀范数增长特性。
摘要：
我们研究了在单位欧几里得球$\mathbb{S}^q$上，由最小二乘多项式提供的均匀逼近，其中$q\ge 2$。如同其他多项式投影，研究关注随着次数$n$趋于无穷时，相关勒贝格常数的增长情况，即最小二乘算子的均匀范数。如果次数为$n$的最小二乘多项式基于一组点，这些点是正权数求积规则的节点，精确度为$2n$，则我们提出两种不同的充分条件，以确保勒贝格常数以最小投影阶随$n$增长。因此，在我们的假设下，最小二乘和超插值多项式在均匀范数方面具有可比的逼近效果。
英文摘要：
We investigate the uniform approximation provided by least squares polynomials on the unit Euclidean sphere $\mathbb{S}^q$ in $\mathbb{R}^{q+1}$, with $q\ge 2$. Like any other polynomial projection, the study concerns the growth, as the degree $n$ tends to infinity, of the associated Lebesgue constant, i.e., of the uniform norm of the least squares operator. If the least squares polynomial of degree $n$ is based on a set of points, which are nodes of a positive weighted quadrature rule of degree of exactness $2n$, then we state two different sufficient conditions for having an optimal Lebesgue constant that increases with $n$ at the minimal projections order. Hence, under our assumptions least squares and hyperinterpolation polynomials provide a comparable approximation with respect to the uniform norm.
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