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Properties of discrete non-multiplicative operators

Properties of discrete non-multiplicative operators The paper is focused on general sequences of discrete linear operators, say $$(L_n)_{n\ge 1}$$ ( L n ) n ≥ 1 . The special case of positive operators is also to our attention. Concerning the quantity $${\Delta } (L_n,f,g):=L_n(fg)-(L_n f)(L_n g), f$$ Δ ( L n , f , g ) : = L n ( f g ) - ( 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 $${\Delta }(L_n,f,g)$$ Δ ( L n , f , g ) . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Analysis and Mathematical Physics Springer Journals

Properties of discrete non-multiplicative operators

Analysis and Mathematical Physics , Volume 9 (1) – Aug 22, 2017

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer International Publishing AG
Subject
Mathematics; Analysis; Mathematical Methods in Physics
ISSN
1664-2368
eISSN
1664-235X
DOI
10.1007/s13324-017-0186-4
Publisher site
See Article on Publisher Site

Abstract

The paper is focused on general sequences of discrete linear operators, say $$(L_n)_{n\ge 1}$$ ( L n ) n ≥ 1 . The special case of positive operators is also to our attention. Concerning the quantity $${\Delta } (L_n,f,g):=L_n(fg)-(L_n f)(L_n g), f$$ Δ ( L n , f , g ) : = L n ( f g ) - ( 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 $${\Delta }(L_n,f,g)$$ Δ ( L n , f , g ) .

Journal

Analysis and Mathematical PhysicsSpringer Journals

Published: Aug 22, 2017

References