Abstract:
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 $L^p$.
Keywords:Bernstein-type inequality, Jackson–Nikol'skii inequality, Dirichlet kernel, trigonometric polynomial, spectrum of a polynomial, hyperbolic cross.