On approximation properties of a family of linear operators at critical value of parameter

We introduce the family of linear operatorsAαfx=1Γα∫0∞tα-1Stfxdt,α>0associated to a certain “admissible bunch” of operators StSt, t>0t>0, acting on Lp(Rn,dmLp(Rn,dm, and investigate the approximation properties of this family as α→0+α→0+. We give some applications to the Riesz and the Bessel potentials generated by the ordinary (Euclidean) and generalized translations.