Geodesic
On Bernstein inequalities for multivariate trigonometric polynomials in $L_{p}$, $0\leq p\leq \infty $
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 2, pp. 449-459
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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 $n$.
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 $n$.
DOI : 10.21136/CMJ.2021.0531-20
Classification : 41A10, 41A17
Keywords: univariate trigonometric polynomial; multivariate trigonometric polynomial; multivariate algebraic polynomial; Bernstein inequality; $L_{p}$-norm 
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Zhu, Laiyi; Zhao, Xingjun. On Bernstein inequalities for multivariate trigonometric polynomials in $L_{p}$, $0\leq p\leq \infty $. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 2, pp. 449-459. doi: 10.21136/CMJ.2021.0531-20

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