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I have read this post (and similar ones):

Decay of the Fourier transform of a non-differentiable function

and I have the following doubt: If

  1. $f\in C^{n}(\mathbb{R})$
  2. $f^{(n+1)}$ is piecewise and of bounded variation
  3. All $f^{(k)}$ are integrable functions for $k=0, \ldots, n+1$

Can I conclude that $\hat{f}(\omega)= o(\omega^{-n-2}), \:\omega \to + \infty.$ I just need a reference to find this precise or a similar statement.

Thanks for your help.

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    $\begingroup$ Just integrate by parts $n+2$ times. Derivative of a function of bounded variation is a finite (signed) measure. Fourier transform of a measure is bounded. $\endgroup$
    – Alexandre Eremenko
    Commented Dec 23, 2022 at 14:47
  • $\begingroup$ Thanks for your help. $\endgroup$
    – Benigno
    Commented Dec 24, 2022 at 10:51

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