In this paper, we discuss the properties of the generalized translation operator generated by the system of functionssuperscriptsubscript2𝑘1𝜋2𝑡𝑘1 in the spaces superscript𝐿𝑝01, 𝑝1. The translation operator is applied to the study of the Nikol’skii inequality between the uniform norm and the superscript𝐿𝑝-norm of polynomials in this system. © 2022, Pleiades Publishing, Ltd.