On Bernstein inequalities for multivariate trigonometric polynomials in \(L_p\), \(0\leq p\leq\infty\). (English) Zbl 07547214

Summary: Let \(\mathbb{T}_n\) be the space of all trigonometric polynomials of degree not greater than \(n\) with complex coefficients. Arestov extended the result of Bernstein and others and proved that \(\|(1/n)T'_{n}\|_{p}\leq\| T_{n}\|_{p}\) for \(0\leq p\leq\infty\) and \(T_n\in\mathbb{T}_n\). We derive the multivariate version of the result of Golitschek and Lorentz \[\Bigl\|\Bigl |T_{n}\cos\alpha +\frac{1}{n}\nabla T_{n}\sin\alpha\Bigr|_{l_{\infty}^{(m)}}\Bigr\|_{p}\leq\| T_{n}\|_{p},\quad 0\leq p\leq\infty\] for all trigonometric polynomials (with complex coeffcients) in \(m\) variables of degree at most \(n\).


41A10 Approximation by polynomials
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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