Abstract
It is proved in this short note that for any polynomial p of d variables and degree at most n we have the sharp Bernstein–Markov type inequality ∫Bd(1−|x|2)μ+1|∂p|2≤M∫Bd(1−|x|2)μ|p|2,μ>−1, with M=n(n+d+2μ) and M=n(n+d+2μ)−d+1 if n is even or odd, respectively. Here Bd is the unit ball in Rd and ∂p stands for the gradient of polynomial. The sharp upper bounds are attained for certain Jacobi type polynomials.
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