In the paper we prove asymptotic estimates for sequences of linear positive operators on the space \(C_{2\pi }\left( \mathbb {R}^{k}\right) \) . A sample result: Let \(K_{n}:\mathbb {R}\rightarrow \left[ 0,\infty \right) \) be a sequence of even continuous \(2\pi \) -periodic functions such that \(\frac{1}{ 2\pi }\int _{-\pi }^{\pi }K_{n}\left( t\right) dt=1\) for all \(n\in \mathbb {N}\) and let \(V_{n}:C_{2\pi }\left( \mathbb {R}^{k}\right) \rightarrow C_{2\pi }\left( \mathbb {R}^{k}\right) \) be the operator defined by \(\begin{aligned} & V_{n}\left( f\right) \left( x_{1},...,x_{k}\right) =\frac{1}{\left( 2\pi \right) ^{k}}\int _{\left[ -\pi ,\pi \right] ^{k}}f\left( x_{1}-t_{1},...,x_{k}-t_{k}\right) \\ & \quad K_{n}\left( t_{1}\right) \cdot \cdot \cdot K_{n}\left( t_{k}\right) dt_{1}\cdot \cdot \cdot dt_{k}. \end{aligned}\) Then the following assertions are equivalent: (i) \(\lim \nolimits _{n\rightarrow \infty }\frac{1-\rho _{n,2}}{1-\rho _{n,1}}=4\) ; \(\rho _{n,1}=\frac{1}{2\pi }\int _{-\pi }^{\pi }K_{n}\left( t\right) \cos tdt\) , \(\rho _{n,2}=\frac{1}{2\pi }\int _{-\pi }^{\pi }K_{n}\left( t\right) \cos 2tdt\) . (ii) For every \(x\in \mathbb {R}^{k}\) and every \(f\in C_{2\pi }\left( \mathbb { R}^{k}\right) \) twice differentiable at x we have \(\lim \nolimits _{n \rightarrow \infty }\frac{V_{n}\left( f\right) \left( x\right) -f\left( x\right) }{1-\rho _{n,1}}=\Delta f\left( x\right) \) , where \(\Delta \) is the Laplacian.
