Properties of discrete non-multiplicative operators

The paper is focused on general sequences of discrete linear operators, say \((L_n)_{n\ge 1}\). The special case of positive operators is also to our attention. Concerning the quantity \({\Delta } (L_n,f,g):=L_n(fg)-(L_n f)(L_n g), f\) and g belonging to some certain spaces, we propose different estimates. Firstly, we study its asymptotic behavior in Voronovskaja’s sense. Examples are presented. Secondly, we prove an extension of Chebyshev–Gruss type inequality for the above quantity. Special cases are investigated separately. Finally we establish sufficient conditions that ensure statistical convergence of the sequence \({\Delta }(L_n,f,g)\).