On the quadrature exactness in hyperinterpolation

This paper investigates the role of quadrature exactness in approximation scheme hyperinterpolation. Constructing a hyperinterpolant degree $n$ requires positive-weight rule with $2n$. We examine behavior such when required $2n$ is relaxed to $n+k$ $0<k\leq n$. Aided by Marcinkiewicz--Zygmund inequality, we affirm that $L^2$ norm exactness-relaxing hyperinterpolation operator bounded constant independent $n$, and this convergent as $n\rightarrow\infty$ if $k$ positively correlated $n$. Thus, family candidate rules for constructing hyperinterpolants can be significantly enriched, number points considerably reduced. As potential cost, relaxation may slow convergence rate terms reduced degrees exactness. Our theoretical results are asserted numerical experiments on three best-known rules: Gauss quadrature, Clenshaw--Curtis spherical $t$-designs.