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First of all, I am beginner for Fourier series I am trying to learn it.

I have a function $f(x) \in L^2 \left( \left[\pi,\pi \right] \right) $ and its Fourier Series is $$\sum ^{\infty }_{n=0}\dfrac{1}{n^{3}}\sin nx$$

My question is how can I say $f$ is periodic and continuous.

$\bullet$ for periodic part, I guess it is from definition. A Fourier series is an expansion of a periodic function $f$ in terms of an infinite sum of sines and cosines.

Any help would be appreciated.

Bumblebee
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Fuat Ray
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  • First of all, you should explain why $f$ is well-defined. After that, what theorems do you know about the continuity of the limit of a sequence of functions that converges uniformly? And why this series converges uniformly? – mathcounterexamples.net Apr 06 '23 at 09:00

1 Answers1

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The terms are dominated by those of $\sum \frac 1 {n^{3}}$ so the Fourier series converges uniformly to some function $g$ which is continuous since uniform limits if continuous functions are continuous. The partial sums converge to $f$ in $L^{2}$ and to $g$ pointwise which implies $f=g$ a.e. Hence, $f$ is almost everywhere equal to a continuous function.

geetha290krm
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