Rate of convergence of beta operators of second kind for functions with derivatives of bounded variation

Obviously the operators Ln are positive linear operators on the space of locally integrable functions on I of polynomial growth as t→∞, provided that n is sufficiently large. In 1995, Stancu [10] gave a derivation of these operators and investigated their approximation properties. We mention that similar operators arise in the work by Adell et al. [3, 4] by taking the probability density of the inverse beta distribution with parameters nx and n. Recently, Abel [1] derived the complete asymptotic expansion for the sequence of operators (1.1). In [2], Abel and Gupta studied the rate of convergence for functions of bounded variation. In the present paper, the study of operators (1.1) will be continued. We estimate their rate of convergence by the decomposition technique for absolutely continuous functions f of polynomial growth as t→ +∞, having a derivative f ′ coinciding a.e. with a function which is of bounded variation on each finite subinterval of I . Several researchers have studied the rate of approximation for functions with derivatives of bounded variation. We mention the work of Bojanić and Chêng (see [5, 6]) who estimated the rate of convergence with derivatives of bounded variation for Bernstein and Hermite-Fejer polynomials by using different methods. Further papers on the subject were written by Bojanić and Khan [7] and by Pych-Taberska [9]. See also the very recent paper by Gupta et al. [8] on general class of summation-integral type operators.