Abstract
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–Grüss type inequality for the above quantity. Special cases are investigated separately. Finally we establish sufficient conditions that ensure statistical convergence of the sequence \({\Delta }(L_n,f,g)\).
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The author is thankful to the referee who carefully checked the manuscript. The comments led to several improvements.
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The author Agratini Octavian declares that he has no conflict of interest.
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Agratini, O. Properties of discrete non-multiplicative operators. Anal.Math.Phys. 9, 131–141 (2019). https://doi.org/10.1007/s13324-017-0186-4
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DOI: https://doi.org/10.1007/s13324-017-0186-4
Keywords
- Linear operator
- Voronovskaja formula
- Grüss-type inequality
- Bernstein operator
- Jain operator
- Generalized sample operator
- Statistical convergence