Abstract:
We establish for $0<p<1$ the analog of the Bernstein–Zygmund inequality for the derivative
of a trigonometric polynomial
$$
\int_{-\pi}^\pi|t_n'(x)|^pdx\leqslant c_pn^p\int_{-\pi}^\pi|t_n(x)|^pdx.
$$
We prove weighted inequalities, exact in the sense of order, for trigonometric polynomials
and their derivatives in various integral metrics with exponents $0<p$, $q\leqslant\infty$.
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