A Generalized Translation Operator Generated by the Sinc Function on an Interval
Vitalii Vladimirovich Arestov, M. V. Deikalova · Proceedings of the Steklov Institute of Mathematics · 2023
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 $$\mathfrak{S}$$ .