  We prove estimates on the Lebesgue constants of the hyperinterpolation
operator for functions on the unit ball $B^d \subset \RR^d$, with respect to
Gegenbauer weight functions, $(1-|\xb|^2)^{\mu-1/2}$. The relationship between
orthogonal polynomials on the sphere and ball is exploited to achieve this
result, which provides an improvement on known estimates of the Lebesgue
constant for hyperinterpolation operators on $B^2$.
