Geodesic
A Generalized Translation Operator Generated by the Sinc Function on an Interval
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 4, pp. 27-48

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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 $\mathfrak{S}$.
Keywords: generalized translation, sinc function, inequality of different metrics. 
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V. V. Arestov; M. V. Deikalova. A Generalized Translation Operator Generated by the Sinc Function on an Interval. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 29 (2023) no. 4, pp. 27-48. http://geodesic.mathdoc.fr/item/TIMM_2023_29_4_a2/