Hi! I am Max.

Question:
We know that
if a series converges absolutely at magnitude $r$,
then it converges at every $z$ such that $|z|=r$.
Is the converse true?
My answer: abs_converse.pdf

$$ \hat{f}(\xi) = \int_{-\infty}^\infty f(x) e^{-2\pi ix\xi} dx = \int_{-\infty}^\infty f(-x) e^{-2\pi i(-x)\xi} dx = \pm \int_{-\infty}^\infty f(x) e^{-2\pi ix(-\xi)} dx = \pm \hat{f}(-\xi) $$ This proves the $\implies$ direction.
Similar argument but with Fourier inversion formula proves the $\impliedby$ direction.

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