Geodesic
Inequalities of Bernstein and Jackson--Nikol'skii Type
Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 95-104

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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. 
@article{MZM_2006_80_1_a11,
     author = {M. B. Sikhov},
     title = {Inequalities of {Bernstein} and {Jackson--Nikol'skii} {Type}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {95--104},
     publisher = {mathdoc},
     volume = {80},
     number = {1},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a11/}
}
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M. B. Sikhov. Inequalities of Bernstein and Jackson--Nikol'skii Type. Matematičeskie zametki, Tome 80 (2006) no. 1, pp. 95-104. http://geodesic.mathdoc.fr/item/MZM_2006_80_1_a11/