Direct and inverse theorems for Szász-Lupas type operators in simultaneous approximation. (English) Zbl 1119.41017

The authors introduce a new modification of Szász–Lupas integral operators which can be used in approximation of real functions, in the form \[ (V_nf)(x)=(n-1)\sum_{k=0}^\infty s_{n,k}(x)\int_0^\infty p_{n,k}(t)f(t)\,dt, \] where \(f\in C[0,\infty)\), \(s_{n,k}(x)=\frac{e^{-nx}(nx)^k}{k!}\), \(p_{n,k}(x)=\binom{n+k-1}kx^k(1+x)^{-n-k}\). After proving some auxiliary results, they deduce direct and inverse theorems on simultaneous approximations of continuous functions by operators of the given form.


41A35 Approximation by operators (in particular, by integral operators)
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