Optimal Lebesgue constants for least squares polynomial approximation on the (hyper)sphere
This provides theoretical guarantees for the uniform approximation quality of least squares polynomials on the sphere, benefiting numerical analysis and approximation theory.
The paper investigates the uniform approximation error of least squares polynomial approximation on the unit sphere, showing that under certain conditions on the sampling points (nodes of a positive weighted quadrature rule of exactness 2n), the Lebesgue constant grows at the minimal order, making least squares and hyperinterpolation polynomials comparable in uniform norm.
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.