Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in $\mathbb{R}^{d}$

We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions

on the standard simplex $\Sigma ^{d}$ in $\mathbb{R}^{d}$. It is proved that such an expansion is uniformly $(C, \delta )$ summable on the simplex for any continuous function if and only if $\delta > |\alpha |_{1} + (d-1)/2$. Moreover, it is shown that $(C, |\alpha |_{1} + (d+1)/2)$ means define a positive linear polynomial identity, and the index is sharp in the sense that $(C,\delta )$ means are not positive for $0 <\delta <|\alpha |_{1} + (d+1)/2$.