Summability of Fourier Orthogonal Series for Jacobi Weight Functions on the Simplex in R

We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions x α1−1/2 1 · · · xαd−1/2 d (1 − |x|1)αd+1−1/2 on the standard simplex Σd in Rd. It is proved that such an expansion is uniformly (C, δ) summable on the simplex for any continuous function if and only if δ > |α|1 + (d − 1)/2. Moreover, it is shown that (C, |α|1 + (d + 1)/2) means define a positive linear polynomial identity, and the index is sharp in the sense that (C, δ) means are not positive for 0 < δ < |α|1 + (d + 1)/2.