Quantitative Voronovskaya-Type Theorems for Fejér-Korovkin Operators

Year 2020, , 150 - 164, 01.12.2020

Jorge Bustamante , Lázaro Flores De Jesús

https://doi.org/10.33205/cma.818715

Cited By: 2

Abstract

In recent times quantitative Voronovskaya type theorems have been presented in
spaces of non-periodic continuous functions. In this work we proved similar results
but for Fejér-Korovkin trigonometric operators. That is we measure the rate of convergence
in the associated Voronovskaya type theotem. Recall that these operators provide the optimal rate in approximation
by positive linear operators. For the proofs we present new
inequalities related with trigonometric polynomials as well as with the convergence factor
of the Fej\'er-Korovkin operators. Our approach includes spaces of
Lebesgue integrable functions.

Keywords

quantitative Voronovskaya type theorems, positive linear operator, Fejér-Korovkin operators

Supporting Institution

I have not received support.

References

• J. Bustamante, L. Flores-de-Jesús: Strong converse inequalities and quantitative Voronovskaya-type theorems for trigonometric Fejér sums. Constr. Math. Anal. 3 (2) (2020), 53-63.
• P. L. Butzer, E. Gorlich: Saturationsklassen und asymptotische Eigenschaften trigonometrischer singulärer Integrale. (German), 1966 Festschr. Gedächtnisfeier K.Weierstrass, Westdeutscher Verlag, Cologne, 339–392.
• P. L. Butzer, R. J. Nessel: Fourier Analysis and Approximation. New York-Base1 (1971).
• P. L. Butzer, E. L. Stark: On a trigonometric convolution operator with kernels having two zeros of simple multiplicity. Acta Math. Acad. Sci. Hung. 20 (1969), 451-461.
• S. Foucart, Y. Kryakin and A. Shadrin: On the exact constant in the Jackson-Stechkin inequality for the uniform metric. Constr. Approx. 29 (2009), 157-179.
• P. P. Korovkin: An asymptotic property of positive methods of summation of Fourier series and best approximation of functions of class Z2 by linear positive polynomial operators. (in Russian), Uspehi Mat. Nauk 6 (84) (1958), 99-103.
• I. M. Petrov: Order of approximation of functions of the class Z for some polynomial operators. (in Russian), Uspehi Mat. Nauk 13 (84) (1958), 127-131.
• E. L. Stark: The kernel of Fejér-Korovkin: a basic tool in the constructive theory of functions. Functions, series, operators, Vol. I, II (Budapest, 1980), Colloq. Math. Soc. János Bolyai, 35, North-Holland, Amsterdam, 1983, 1095-1123.
• A. F. Timan: Theory of Approximation of Functions of Real Variable. Pergamon Press, 1963.
• A. Zygmund: Trigonometric series. Third Edition, Vol I and II combined, Cambridge Mathematical Library, 2002.