On Voronovskaja formula for linear combinations of Mellin–Gauss–Weierstrass operators

Here we give a Voronovskaja formula of high order for linear combinations of the convolution operators(Gw,rf)(s)=∫0+∞∑j=1rαjKjw(t)f(st)dtt,where KwKw is the Mellin–Gauss–Weierstrass kernel. This kind of operator provides a better order of pointwise approximation and leads to asymptotic formulae of typelimw→+∞wν[(Gw,rf)(s)-f(s)]=A(f,ν)where r,ν∈Nr,ν∈N and A(f,ν)A(f,ν) is a differential operator containing the derivatives of f up to order ν.