On the norms and minimal properties of de la Vallée Poussin’s type operators

Abstract

In this paper we consider the de la Vallee Poussin’s type operators \(H_{rn,sn}\) $$\begin{aligned} H_{rn,sn}:=\frac{F_{rn}+F_{rn+1}+...+F_{sn-1}}{(s-r)n}, \end{aligned}$$ where \(F_k\) are classical Fourier projections onto \(\varPi _k\) (the space of trigonometric polynomials of degree less than or equal to k). We determine when \(H_{n,sn}\) is the minimal generalized projection and provide the asymptotic behavior of the norm \(\Vert H_{n,sn}\Vert \). Additionally, we contrast the results obtained for the trigonometric system to the results obtained for the Rademacher system.