On hyperinterpolation on the unit ball

We prove estimates on the Lebesgue constants of the hyperinterpolation operator for functions on the unit ball $B^d\subset {\text{RR}}^d$, with respect to Gegenbauer weight functions, $(1-\mathrm{\mid }\text{xb}{\mathrm{\mid }}^2{)}^{\mu -1\mathrm{/}2}$. 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 $B^2$.