Inequalities of Bernstein and Jackson–Nikol'skii Type and Estimates of the Norms of Derivatives of Dirichlet Kernels

We obtain Bernstein and Jackson–Nikol'skii inequalities for trigonometric polynomials with spectrum generated by the level surfaces of a function $\Lambda(t)$, and their sharpness is studied under a specific choice of $\Lambda(t)$. Estimates of the norms of derivatives of Dirichlet kernels with harmonics generated by the level surfaces of the function $\Lambda(t)$ are established in $L^p$.