Convergence of the Vall\'ee--Poussin means for Fourier--Jacobi sums

I. I. Sharapudinov ; I. A. Vagabov

Matematičeskie zametki, Tome 60 (1996) no. 4, pp. 569-586

Résumé

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 $0$, 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 $C[-1,1]$.