Strong converse inequality for weighted approximation of functions by the Szász-Mirakjan-Kantorovich operator. (English) Zbl 07845563
The classical Szasz-Mirakjan operator is defined in [G. Mirakyan, C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 31, 201–205 (1941; Zbl 0025.04002)] and [O. Szasz, J. Res. Natl. Bur. Stand. 45, No. 3, 239–245 (1950; Zbl 1467.41005)] for bounded functions \(f(x)\) in \([0,\infty ) \) by the formula \[S_{n} f(x)= S_{n } (f, x)= \sum_{k=0}^{\infty} f(\frac{k}{n} ). s_{n, k} (x) , x \geq 0 \] where \( s_{n, k} (x) = e^{-nx} \frac{(nx)^{k}}{k!}.\) It was introduced in order to approximate continuous functions in uniform norm and it is not suitable for approximation of functions in \(L_{p}\) spaces. There are many modifications of the classical Szasz-Mirakjan operator. The most important are the Durrmeyer type (see [H. S. Kasana et al., KFAS Proc. Ser. 3, 29–41 (1988; Zbl 0717.41041)] or [S. M. Mazhar and V. Totik, Acta Sci. Math. 49, 257–269 (1985; Zbl 0611.41013)]) and Kantorovich type (see [Z. Ditzian and V. Totik, Moduli of smoothness. New York etc.: Springer (1987; Zbl 0666.41001)]) modifications.
In this paper, the authors investigate the weighted approximation of functions in \(L_{p}\) norm by Kantorovich modification of Szasz-Mirakjan operator with weights of type \( (1 + x)^ \alpha\), \(\alpha \in {R}\). By using an appropriate \(K\)-functional they prove a strong converse inequality for the weighted error of approximation and characterize it. They also prove a Voronovskaya and Bernstein-type inequalities for the Szasz-Mirakjan-Kantorovich operator.
In this paper, the authors investigate the weighted approximation of functions in \(L_{p}\) norm by Kantorovich modification of Szasz-Mirakjan operator with weights of type \( (1 + x)^ \alpha\), \(\alpha \in {R}\). By using an appropriate \(K\)-functional they prove a strong converse inequality for the weighted error of approximation and characterize it. They also prove a Voronovskaya and Bernstein-type inequalities for the Szasz-Mirakjan-Kantorovich operator.
Reviewer: Hüseyin Çakallı (İstanbul)
MSC:
| 41A36 | Approximation by positive operators |
| 41A25 | Rate of convergence, degree of approximation |
| 41A27 | Inverse theorems in approximation theory |
| 41A17 | Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) |
Keywords:
Szasz-Mirakjan operator; Kantorovich operator; K-functional; weighted approximation; strong converse inequalityReferences:
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