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 \(L^{q}=L^{q}((0,1),{\upsilon})\), \(q\geq 1\), 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 \(L^{q}\), \(1\leq q\leq\infty\). 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 \(\mathfrak{S}\).