On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu
DOI:
https://doi.org/10.33993/jnaat301-687Abstract
In 1931, Tiberiu Popoviciu has initiated a procedure for the construction of sequences of linear positive operators of approximation. By using the theory of polynomials of binomial type \((p_m)\) he has associated to a function \(f\in C[0,1]\) a linear operator defined by the formula
\[
\left( T_m f\right) (x) = \tfrac{1}{p_m(1)} \textstyle\sum\limits _{k=0} ^m \tbinom{m}{k}
p_k (x) p_{m-k} (1-x) f\big(\tfrac{k}{m}\big).
\]
Examples of such operators were considered in several subsequent papers.
In this paper we present a convergence theorem corresponding to the sequence \(\left( T_mf\right)\) and we also present a more general sequence of operators of approximation \(S_{m,r,s}\), where \(r\) and \(s\) are nonnegative integers such that \(2sr\leq m\).
We give an integral expression for the remainders, as well as a representation by using divided differences of second order.
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O. Agratini, Application of divided differences to the study of monotonicity of a sequence of D. D. Stancu polynomials, Rev. Anal. Numér. Theor. Approx., 25 (1996), 3-10.
W. Z. Chen, Approximation theorems for Stancu-type operators, Xiamen Daxue, 32 (1993), 679-684.
B. Della Vecchia, On the approximation of functions by means of the operators of D. D. Stancu, Studia Univ. Babeş-Bolyai, 37 (1992), 3-36.
I. Horova and M. Budikova, A note on D. D. Stancu operators, Ricerche di Matematica, 44 (1995), 397-407.
L. Lupaş and A. Lupaş, Polynomials of binomial type and approximation operators, Studia Univ. Babeş-Bolyai, 32 (1987), 61-69.
C. Manole, Approximation operators of binomial type, Res. Seminar on Numerical and Statistical Calculus (Cluj), Preprint Nr. 9 (1987), 93-98.
G. Mastroianni and M. R. Occorsio, Sulle derivate dei polinomi di Stancu, Rend. Accad. Sci. M.F.N., 45 (1978), 273-281.
G. Moldovan, Discrete convolutions and linear positive operators, Ann. Univ. Sci. Budap., 15 (1972), 31-44.
T. Popoviciu, Remarques sur les polynomes binomiaux, Bul. Soc. Sci. Cluj, 6 (1931), 146-148.
T. Popoviciu, Sur le reste dans certaines formules lineaires d'approximation de l'analyse, Mathematica, 1 (24) (1959), 95-142.
P. Sablonnière, Positive Bernstein-Sheffer operators, J. Approx. Theory, 83 (1995), 330-341, https://doi.org/10.1006/jath.1995.1124 DOI: https://doi.org/10.1006/jath.1995.1124
D. D. Stancu, The remainder of certain linear approximation formulas, J. SIAM Numer. Anal. B, 1 (1964), 137-163, https://doi.org/10.1137/0701013 DOI: https://doi.org/10.1137/0701013
D. D. Stancu, Approximation of functions by a new class of linear polynomial operators, Rev. Roum. Math. Pures Appl., 13 (1968), 1173-1194.
D. D. Stancu, Approximation of functions by means of a new generalized Bernstein operator, Calcolo, 15 (1983), 211-229, https://doi.org/10.1007/bf02575593 DOI: https://doi.org/10.1007/BF02575593
D. D. Stancu, A note on the remainder in a polynomial approximation formula, Studia Univ. Babeş-Bolyai, 41 (1996), 95-101.
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