A Generalized Translation Operator Generated by the Sinc Function on an Interval

We discuss the properties of the generalized translation operator generated by the system of functions $\mathfrak{S}=\{{(\sin k\pi x)}/{(k\pi x)}\}_{k=1}^\infty$ in the spaces $L^q=L^q((0,1),{\upsilon})$, $q\ge 1$, on the interval $(0,1)$ with the weight $\upsilon(x)=x^2$. We find an integral representation of this operator and study its norm in the spaces $L^q$, $1\le q\le\infty$. The translation operator is applied to the study of Nikol'skii's inequality between the uniform norm and the $L^q$-norm of polynomials in the system $\mathfrak{S}$.