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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.

MSC:

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