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Keywords:
univariate trigonometric polynomial; multivariate trigonometric polynomial; multivariate algebraic polynomial; Bernstein inequality; $L_{p}$-norm
univariate trigonometric polynomial; multivariate trigonometric polynomial; multivariate algebraic polynomial; Bernstein inequality; $L_{p}$-norm
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 $n$.
References:
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