Abstract:
In this paper, we discuss the properties of the generalized translation operator generated by the system of functions
$\left\{ \cos\left(\frac{(2k-1)\pi }{2}t\right)\right\}_{k=1}^\infty$ in the spaces $L^p(0,1)$, $p\ge 1$. The translation operator is applied to the study of the Nikol'skii inequality between the uniform norm and the $L^p$-norm of polynomials in this system.
Keywords:generalized translation operator, trigonometric polynomial, inequality of different metrics.
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