Strong converse inequality for Szász-Durrmeyer operators
For the Szász-Durrmeyer operators \[ L_n(f,x)= n\sum^\infty_{k=0} s_{n,k}(x) \int^\infty_0 s_{n,k}(t)f(t)dt,\quad s_{n,k}(x)= e^{-nx}{(nx)^k\over k!} \] it is proved that for some \(m\), \[ \omega^2_\varphi(f,n^{-1/2})_p\leq M\{|L_n(f,x)- f|_p+ |L_{mn}(f,x)- f|_p\}, \] where \(\varphi(x)^2= x\), \(M>0\) and \(\omega^2_\varphi(f, t)_p\) denotes the second-order Ditzian-Totik modulus of smoothness.
scientific article; zbMATH DE number 947006