On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu

Authors

  • Dimitrie D. Stancu Babeş-Bolyai University, Cluj-Napoca

Abstract

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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References

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Published

2001-02-01

How to Cite

Stancu, D. D. (2001). On the approximation of functions by means of the operators of binomial type of Tiberiu Popoviciu. Rev. Anal. Numér. Théor. Approx., 30(1), 95–105. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/2001-vol30-no1-art13

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