  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.
