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Suppose \(m, n\in \mathbb{N}\), \(p_1,\ldots ,p_n\) are polynomials of degree at most \(m-1\) and \(\lambda_1,\ldots ,\lambda_n\in \mathbb{R}\). Let also \(I\subseteq \mathbb{R}\) be a bounded interval and \(S\subseteq I\) a measurable subset with positive measure. Then

\[ \sup_{t\in I} \lvert r(t)\rvert \le \biggl(\frac{316\lvert I\rvert}{\lvert S\rvert}\biggr)^{nm-1}\sup_{t\in S} \lvert r(t)\rvert. \]
Proof
This theorem is proved by Kovrijkine in [1, Lemma 3]. However, the constant was not specified there. For instance, it appears in [2, Theorem 4.3]
Remarks

See also Link to heading

References Link to heading

  1. O. Kovrijkine, Some estimates of Fourier transforms, Ph.D. thesis, United States -- California: California Institute of Technology, 2000.
  2. M. Egidi and I. Veselić, Scale-free Unique Continuation Estimates and Logvinenko–Sereda Theorems on the Torus, Annales Henri Poincaré, vol. 21, no. 12, pp. 3757–3790, 2020. doi:10.1007/s00023-020-00957-7