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for a periodic function f(x)(continuous & have continuous derivatives), there is a Fourier series for it, i just wonder under what condition f(x) will have finite(or infinite) Fourier-coefficent, which means existing N so kn=zero for any n>N.

and is there any significant difference in function shape between the functions of infinite and infinite spectrum?

Aerterliusi
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  • I suppose a discontinuous f(x) cannot have a finite Fourier series (periodic step function), or a function with one or more points where it is not differentiable (sawtooth function). – Paul Feb 13 '23 at 16:33
  • @Paul yes, because Fourier series is smooth(so i edit my question to restrict function into continuous & have continuous derivatives) – Aerterliusi Feb 13 '23 at 17:43

1 Answers1

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$f(t)=\sum_{n=-N}^N c_n e^{i n t}$ iff $f$ is $2\pi$-periodic, smooth, and $D^{2N} (e^{iNt} f)=0$ where $D (g)(t) =e^{-it} g'(t) $.

reuns
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