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

Abstract:

We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions \[ x_{1}^{\alpha _{1} -1/2} \cdots x_{d}^{\alpha _{d} -1/2}(1-|\mathbf {x}|_{1})^{\alpha _{d+1}-1/2}\] 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 $0 <\delta <|\alpha |_{1} + (d+1)/2$.