×

On the beta approximating operators of second kind. (English) Zbl 0856.41019

The author continues his earlier investigations [Rev. Roum. Math. Pur. Appl. 14, 673-691 (1969; Zbl 0187.32502)] on the use of probabilistic methods for obtaining positive linear operators useful in the constructive theory of functions.
The main result of the paper consists in introducing and investigating the approximation properties of a new beta operator of second kind \(L_m= T_{mx, m+ 1}\), defined by the formula \[ (L_m f)(x):= {1\over B(mx, m+ 1)} \int^\infty_0 f(t) {t^{mx- 1}dt\over (1+ t)^{mx+ m+ 1}}, \] where \(f\) belongs to the linear space of functions \(M[0, \infty)\), defined for \(t\geq 0\), bounded and Lebesgue measurable in each interval \([a, b]\), where \(0< a< b< \infty\). By \(B(p, q)\) one denotes the beta function.
This operator is connected with the probability distribution of Karl Pearson, Type VI. It is distinct from the other beta operators considered earlier by G. Mühlbach, A. Lupaş, R. Upreti, M. K. Khan, J. A. Adell and J. de la Cal.
The paper contains estimations of the orders of approximation, a Voronovskaya type formula and representations for the remainder term.

MSC:

41A35 Approximation by operators (in particular, by integral operators)
41A36 Approximation by positive operators
42A61 Probabilistic methods for one variable harmonic analysis

Citations:

Zbl 0187.32502
PDFBibTeX XMLCite