Let f: R -> R be a 1-periodic function. Prove that F has infinitely many derivatives iff the Fourier?

Question

Coefficients (Cn) of F,
Cn = integral (from 0 to 1)[f(x)e^-2pieinx]dx

do satisfy for any t>0
lim(n->infinity) [n^t[cn]] = 0

Answer 1

you mean (t[cn])^n

if F is infinitily diff, apply the Riemann Lebesgue theorem to F'.

For the other direction the decay condition implies f is its fourier series.