Fourier transform of a real function is real

real-analysis fourier-analysis integration fourier-transform

Solution 1:

The following are equivalent:

  1. $f(-x)=\overline{f(x)}$ for a.e. $x\in\mathbb R$
  2. $\hat f(\xi)\in\mathbb R$ for a.e. $x\in\mathbb R$

The proof is immediate from $$ \overline{\hat f(\xi)} = \int_{\mathbb R} \overline{f(x)}e^{2\pi i x \xi}\,dx = \int_{\mathbb R} \overline{f(-x)}e^{-2\pi i x \xi}\,dx $$

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