# An intermediate Voronovskaja type theorem

• Dumitru Popa
Original Paper

## Abstract

We prove the following intermediate Voronovskaja type theorem: Let $$V_{n}:C \left[ a,b\right] \rightarrow C\left[ a,b\right]$$ be a sequence of positive linear operators such that $$\lim \nolimits _{n\rightarrow \infty }V_{n}\left( f\right) =f$$ uniformly on $$\left[ a,b\right]$$ for every $$f\in C\left[ a,b \right]$$. Let $$x\in \left[ a,b\right]$$ and suppose that there exists a sequence $$\left( \lambda _{n}\right) _{n\in {\mathbb {N}}}\subset \left( 0,\infty \right)$$ such that $$\lim \nolimits _{n\rightarrow \infty }\lambda _{n}=\infty$$ and there exists $$1<p<\infty$$ such that the sequence $$\left( \lambda _{n}^{p}V_{n}\left( \left| \cdot -x\right| ^{p}\right) \left( x\right) \right) _{n\in {\mathbb {N}}}$$ is bounded. Then, for every continuous function $$f:\left[ a,b\right] \rightarrow {\mathbb {R}}$$ differentiable at $$x\in \left[ a,b\right]$$ we have
\begin{aligned} \lim \limits _{n\rightarrow \infty }\lambda _{n}\left[ V_{n}\left( f\right) \left( x\right) -f\left( x\right) V_{n}\left( {\mathbf {1}}\right) \left( x\right) -f^{\prime }\left( x\right) V_{n}\left( \left( \cdot -x\right) \right) \left( x\right) \right] =0. \end{aligned}
As an application we prove a Voronovskaja type theorem for the sequences of the operators $$V_{n}:C\left[ 0,1\right] \rightarrow C\left[ 0,1\right]$$ defined by
\begin{aligned} V_{n}\left( f\right) \left( x\right) =a_{n}b_{n}\int _{0}^{1}\frac{ t^{a_{n}}f\left( \left( 1-t\right) \xi +tx\right) }{t+b_{n}}dt \end{aligned}
where $$\xi \in \left[ 0,1\right]$$, $$\left( a_{n}\right) _{n\in {\mathbb {N}}}$$ is a sequence of natural numbers with $$\lim \nolimits _{n\rightarrow \infty }a_{n}=\infty$$, $$\left( b_{n}\right) _{n\in {\mathbb {N}}}\subset \left( 0,\infty \right)$$ with $$\lim \nolimits _{n\rightarrow \infty }b_{n}=\infty$$ and $$\lim \nolimits _{n\rightarrow \infty }\frac{a_{n}}{b_{n}}\in \left[ 0,\infty \right)$$.

## Keywords

Korovkin approximation theorem Positive linear operators Asymptotic formula Voronovskaia type theorem

## Mathematics Subject Classification

41A35 41A36 41A25

## Notes

### Acknowledgements

We would like to thank the two referees of our paper for carefully reading the manuscript and for such constructive comments, remarks and suggestions which helped improving the first version of the paper.

## References

1. 1.
Altomare, F., Campiti, M.: Korovkin-type Approximation Theory and its Application, vol. 17. de Gruyter Studies in Mathematics, Berlin, New York (1994)Google Scholar
2. 2.
Birou, M.M.: A proof of a conjecture about the asymptotic formula of a Bernstein type operator. Result. Math. 72(3), 1129–1138 (2017)
3. 3.
Cardenas-Morales, D., Garrancho, P., Raşa, I.: Asymptotic formulae via a Korovkin-type result. Abstr. Appl. Anal. 2012, Article ID 217464, 12 p. (2012)Google Scholar
4. 4.
Davis, P.J.: Interpolation and Approximation. Dover Publications, Inc., New York (1975) (Republication, with minor corrections, of the 1963 original, with a new preface and bibliography) Google Scholar
5. 5.
Korovkin, P.P.: On convergence of linear positive operators in the space of continuous functions. Doklady Akademii Nauk 90, 961–964 (1953). (Russian)
6. 6.
Korovkin, P.P.: Linear Operators and Approximation Theory. Hindustan Publ. Corp, India (1960)Google Scholar
7. 7.
Lomeli, H.E., Garcia, C.L.: Variations on a Theorem of Korovkin. Am. Math. Monthly 113, 744–750 (2006)
8. 8.
Nasaireh, F., Raşa, I.: Another look at Voronovskaja type formulas. J. Math. Inequal. 12(1), 95–105 (2018)
9. 9.
Phillips, G.M.: Bernstein polynomials based on the $$q$$-integers. Ann. Numer. Math. 4(1–4), 511–518 (1997)
10. 10.
Sikkema, P.C.: On some linear positive operators. Nederl. Akad. Wet., Proc., Ser. A 73, 327–337 (1970)
11. 11.
Voronovskaja, E.V.: Determination de la forme asymptotique de l’approximation des fonctions par les polynômes de M. Bernstein (Russian), C.R. Acad. Sc. U. R. S. S., 1932, 79–85 (1932)Google Scholar