Convergence of the Vallée–Poussin means for Fourier–Jacobi sums

Let $f\in C[-1,1]$, $-1<\alpha$, $\beta\le0$, $S_n^{\alpha,\beta}(f,x)$ be a partial Fourier–Jacobi sum of order $n$, and let
$$ \begin{aligned} {\mathscr V}_{m,n}^{\alpha,\beta} & ={\mathscr V}_{m,n}^{\alpha,\beta}(f) ={\mathscr V}_{m,n}^{\alpha,\beta}(f,x) \& =\frac 1{n+1}\bigl[S_m^{\alpha,\beta}(f,x)+\dots+S_{m+n}^{\alpha,\beta}(f,x)\bigr] \end{aligned} $$
be the Vallée Poussin means for Fourier–Jacobi sums. It was proved that if $0<a\le m/n\le b$, then there exists a constant $c=c(\alpha,\beta,a,b)$ such that $\|{\mathscr V}_{m,n}^{\alpha,\beta}\|\le c$, where $\|{V}_{m,n}^{\alpha,\beta}\|$ is the norm of the operator ${\mathscr V}_{m,n}^{\alpha,\beta}$ in $C[-1,1]$.