Abstract
In the present paper we study the compositions of the piecewise linear interpolation operator S?n and the Beta-type operator B?n, namely An:= S?n ?B?n and Gn := B?n ? S?n. Voronovskaya type theorems for the operators An and Gn are proved, substantially improving some corresponding known results. The rate of convergence for the iterates of the operators Gn and An is considered. Some estimates of the differences between An, Gn, Bn and S?n, respectively, are given. Also, we study the behaviour of the operators An on the subspace of C[0,1] consisting of all polygonal functions with nodes {0, 1/2,..., n-1/n,1}. Finally, we propose to the readers a conjecture concerning the eigenvalues of the operators An and Gn. If true, this conjecture would emphasize a new and strong relationship between Gn and the classical Bernstein operator Bn.
Highlights
In 1912, Bernstein [2] introduced his famous polynomials in order to prove Weierstrass’ fundamental theorem
The present work is motivated by a problem raised by Lupas and Gonska in 2006. Their question was if there are non-trivial positive linear operators P and Q such that the classical Bernstein operator can be decomposed as Bn = P ◦ Q
Muhlbach [17, 18] and Lupas [14, 13] and piecewise interpolation at equidistant points in [0, 1]. These Beta operators are given for f ∈ C[0, 1], x ∈ [0, 1], by
Summary
In 1912, Bernstein [2] introduced his famous polynomials in order to prove Weierstrass’ fundamental theorem. The present work is motivated by a problem raised by Lupas and Gonska in 2006 Their question was if there are non-trivial positive linear operators P and Q such that the classical Bernstein operator can be decomposed as Bn = P ◦ Q. Muhlbach [17, 18] and Lupas [14, 13] and piecewise interpolation at equidistant points in [0, 1] These Beta operators are given for f ∈ C[0, 1], x ∈ [0, 1], by f (0), Bn(f ; x) =. The piecewise linear interpolation operators S∆n : C[0, 1] → C[0, 1] at the points.
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