Approximation properties of Baskakov-Durrmeyer operators on some Lebesgue-integrable functions

Let (<f>n) j° be a sequence of real functions having the following properties on an interval / c [0; oo) for every n e N, k e No = N U {0}:

For functions / : / R the discrete Baskakov operators [4] are defined formally by

where p n<k(x) = (—1)*$,, (x)xk/k \ . For measurable functions / : / - > / ? theBaskakovDurrmeyer operators are usually defined by

provided that the right side of ( 1 .1 ) makes sense [8]. Some approximation properties of certain operators (1.1) for functions f e C(I) or

/ e L P(I) can be found e.g. in [5], [8]. The rates of their pointwise convergence for functions / of bounded variation on I are estimated e.g. in [2], [3], [7].