#Python 2, 192 185 bytes
D,E,Q=input();J=0;exec"F=lambda u,m,j,i:+(j[i]<=u<j[-~i])if m<1else(u-j[i])/(j[i+m]-j[i])*F(u,~-m,j,i)+(j[-~i+m]-u)/(j[-~i+m]-j[-~i])*F(u,~-m,j,-~i);print F(D,E,Q,J);J+=1;"*(~-len(Q)-E)
A direct implementation of the formula given with recursion in the lambda and iteration using exec. Prompts for input through the command line upon invocation as u,m,[u_0,u_1,u_2...] and outputs each of the len([u_0,u_1,u_2...])-1-u values on a separate line.