\[ \DeclareMathOperator{\supp}{supp} \]

Let \(d\ge 1\) and \(p \in [1,\infty]\). The Bernstein inequality

\begin{equation*} \lVert f^{(\alpha)}\rVert_{L^p(\mathbb{R}^d)} \le (Cb)^{\lvert \alpha\rvert} \lVert f\rVert_{L^p(\mathbb{R})} \end{equation*}

holds for every multi-index \(\alpha\) and for every \(f\in L^p(\mathbb{R}^d)\) with \(\supp \hat{f} \subset [-b,b]^d\) for some \(b>0\), where \(C>0\) is a numerical constant [1, Theorem 11.3.3].

Links Link to heading

• proof

Related Results Link to heading

• Local Bernstein on the compact manifolds
• Bernstein inequalities on the sphere
  • Bernstein inequality on the sphere by Mhaskar
  • Bernstein inequality on the sphere with covariant derivatives
• Magnetic Bernstein inequality
• Bernstein inequality \(\mathbb{Z}^d\)

Implications Link to heading

• Kovrijkines UCP

References Link to heading

1. R. Boas, Entire functions. New York San Francisco London: Academic press, 1954.