Boundedness in $C[-1,1]$ of the de la Vallée-Poussin means for discrete Chebyshev–Fourier sums

Approximation properties of the de la Vallée-Poussin means $v_{m,n}=v_{m,n}(f)=v_{m,n}(f,x)=v_{m,n}(f,x,N)$ of discrete Chebyshev–Fourier sums in the Chebyshev polynomials forming an orthonormal system on the set $\Omega =\bigl \{-1+2j/(N-1)\bigr \}_{j=0}^{N-1}$ with respect to the weight $\rho (x)=2/N$ are considered. For $0<d \leqslant m/n \leqslant b$ and $n \leqslant a\sqrt N$ the existence of a constant $c=c(a,b,d)$ is established such that $\|v_{m,n}\| \leqslant c$, where $\|v_{m,n}\|$ is the norm of the operator $v_{m,n}$ in the space $C[-1,1]$. As a consequence, it is proved for an algebraic polinomial $p_n(x)$) of degree $n \leqslant a\sqrt N$ that if $\max \bigl \{|p_n(x)|:x \in \Omega \bigr \} \leqslant 1$, then the following estimate is valid: $\|p_n\|=\max \bigl \{|p_n(x)|:x\in [-1,1]\bigr \} \leqslant c(a)$.