On Bernstein inequalities for multivariate trigonometric polynomials in L_p, 0 ⩽ p ⩽ ∞

Abstract

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 ${\left\| {(1/n)T_n^\prime } \right\|_p} \leqslant {\left\| {{T_n}} \right\|_p}$ for 0 ⩽ p ⩽ ∞ and ${T_n} \in {{\mathbb{T}}_n}$. We derive the multivariate version of the result of Golitschek and Lorentz

$${\left\| {{{\left| {{T_n}\cos \alpha + {1 \over n}\nabla {T_n}\sin \alpha } \right|}_{l_\infty ^{(m)}}}} \right\|_p} \leqslant{\left\| {{T_n}} \right\|_p},\,\,\,\,\,\,\,0 \leqslant p \leqslant\infty $$

for all trigonometric polynomials (with complex coefficients) in m variables of degree at most n.