We discuss the properties of the generalized translation operator generated by the system of functions S={(sinkπx)/(kπx)}∞k=1 in the spaces Lq=Lq((0,1),υ), q≥1, on the interval (0,1) with the weight υ(x)=x2. We find an integral representation of this operator and study its norm in the spaces Lq, 1≤q≤∞. The translation operator is applied to the study of Nikol'skii's inequality between the uniform norm and the Lq-norm of polynomials in the system S.