The L1 saturation class of the Kantorovičoperator

Saturation of Kantorovich type operators

“…An analogous result for the Kantorovich polynomials was proved byV. Major [4], however his method cannot be directly modified so as to prove our theorem because his proof rested on a delicate identity and the candidate of a test function is not integrable in our case.…”

Kantorovi? operators of second order

“…(k+ 1~ k where ~ as in (4). For the second order difference in (5) Without loss of generality we may assume f>~ 0 a.e., due to Proposition 1.…”

“…P,, is the so-called Kantorovie operator. In the last years approximation by Kantorovie operators was an object of investigation for several authors, e. g. GRUNIgMANN [1], MAIER [4], Mi)LLER [5], RIEMENSCHNEIDER [9]. As is easily seen, B, and P, are connected by the relation After compiling elementary properties of Q, in section 2, in section 3 we shall consider the restriction of Q, onto ~g [0,1].…”

Kantorovich-Bernstein polynomials

“…For the Kantorovich-Bernstein and the Bernstein operators it is the investigation of the class for which Cn = 1In. This was carried out for (1.4) by V. Maier [8] when p = 1 and by S. D. Riemenschneider [9] when 1 <p <or, while for (1.3) the class of functions is known to be x(1 -x)f" r L~[0, 1]. Note that the saturation theorem on Bernstein polynomials by G. G. Lorentz E7, p. 102] yields a somewhat different result, namely A different technique from that used for the saturation result was used to determine the class of functions for which o" a = n -~ by the first author for the Bernstein polynomials (1.3) in E33 and [4], and by V. Totik for the KantorovichBernstein polynomials (1.4) in [10] and [11].…”