Approximation of classes of convolutions by linear operators of special form

A parametric family of operators G _ρ is constructed for the class of convolutions W _p,m (K) whose kernel K was generated by the moment sequence. We obtain a formula for evaluating $E(W_{p,m} (K);G_rho )_p : = mathop {sup }limits_{f in W_{p,m} (K)} left| {f - G_rho (f)} right|_p .$ . For the case in which W _p,m (K)=W _r,β ^p,m , we obtain an expansion in powers of the parameter ?=?ln ρ for E(W _p,m ^r,β ; G _ρ,r ) _p , where β ∈ ?, γ > 0, and m ∈ ?, while p = 1 or p = ∞.