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

Abstract

Letf εC[−1, 1], −1<α,β≤0, let$f \in C[ - 1, 1], - 1< \alpha , \beta \leqslant 0$, letS ^{α, β} _n (f, x) be a partial Fourier-Jacobi sum of ordern, and let

$$\nu _{m, n}^{\alpha , \beta } = \nu _{m, n}^{\alpha , \beta } (f) = \nu _{m, n}^{\alpha , \beta } (f,x) = \frac{1}{{n + 1}}[S_m^{\alpha ,\beta } (f,x) + ... + S_{m + n}^{\alpha ,\beta } (f,x)]$$

be the Vallée-Poussin means for Fourier-Jacobi sums. It was proved that if 0<a≤m/n≤b, then there exists a constantc=c(α, β, a, b) such that ‖ν ^{α, β} _{m, n} ‖ ≤c, where ‖ν ^{α, β} _{m, n} ‖ is the norm of the operator ν ^{α, β} _{m, n} inC[−1,1].