About characterization of one K-functional

We characterize the K-functional K˜ψ(f,t)p (used in approximating functions by Durrmeyer and Kantorovich modifications of Baskakov operator)K˜ψ(f,t)p=inf⁡{‖f−g‖p+t‖D˜g‖p:f−g∈Lp[0,∞),g∈W˜p[0,∞)} where D˜=ddx(x(1+x)ddx),W˜p[0,∞)={f:f,f′∈ACloc(0,∞),D˜f∈Lp[0,∞),limx→0+,∞⁡ψ(x)f′(x)=0} by appropriate moduli of smoothness.