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On the convergence of modified Baskakov operators. (English) Zbl 0969.41013

The Baskakov operators associated with fC[0,) are defined by

V n (f,x):= k=0 P n,k (x)fk n,


P n,k (x):=n+k-1 kx k (1+x) n+k ,

whenever the righthand side exists. The modified Baskakov operators are derived from the original ones by truncating the infinite sum depending on the value of x, namely,

V n,δ (f,x):= k=0 [n(x+δ)] P n,k (x)fk n·

The modified operators become interesting if δ0 as n. The authors prove that if |f(t)|Ae st , 0t<, for some positive A and s, then for any x[0,), V n,δ(n) (f,x)f(x), provided V n (t-x) 2 , xδ -2 (n)0. It is well known that V n (t-x) 2 , xC(x)n -1 and the authors show that their condition is best possible in the sense that if n 1/2 δ(n), then for f in the above class, V n,δ(n) (f,x)0, x[0,).

41A36Approximation by positive operators