In read in Kolmogorov-Fomin's (p. 429 here) that if function $f$ is such that $f^{(k-1)}$ is absolutely continuous on any interval and if $f,...,f^{(k)}\in L_1(-\infty,\infty)$, [...] we get $$F[f^{(k)}](\lambda)=(i\lambda)^k F[f](\lambda)$$where $F[f]$ is the Fourier transform of $f$ defined by $F[f](\lambda):=\int_{\mathbb{R}}f(x)e^{-i\lambda x}d\mu_x$. I think it is implicitly intended that all $f,...,f^{(k-1)}$ are absolutely continuous on any interval. Am I right? This is my opinion because, as it is presented in the text, this equality appears to be a consequence of the fact that if $f\in L_1(-\infty,\infty)$ is absolutely continuous on any interval and if $f'\in L_1(-\infty,\infty)$ then $F[f'](\lambda)=i\lambda F[f]$...
Does anybody know more about the fact and can confirm or refute what I think? Thank you so much!