what is it if the Fourier series is $\sum_{n=1}^\infty \frac{\cos 2n\pi x}{n^{s+1}}$

Asked Sep 19 '15 at 08:46

Active Sep 19 '15 at 08:46

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I guess that if $s$ is positive odd integer, then Fourier series $$\sum_{n=1}^\infty \frac{\cos 2n\pi x}{n^{s+1}}$$ is a polynomial of degree $s+1$. For example, if $s=1$, then

$$\sum_{n=1}^\infty \frac{\cos 2n\pi x}{n^2} = c(x^2- x + \frac{1}{6})$$

If $s\geq 0$ is even integer, then Fourier series $$\sum_{n=1}^\infty \frac{\sin 2n\pi x}{n^{s+1}}$$ is a polynomial of degree $s+1$. For example, if $s = 0$, then $$\sum_{n=1}^\infty \frac{\sin 2n\pi x}{n} = c(x-\frac{1}{2})$$

I need your confirmation.

asked Sep 19 '15 at 08:46

Willie Wu

Well, it's the periodic extension of the restriction of a polynomial to $[0,1)$. Prove it by induction on $s$, noting that for $s > 1$, you can differentiate the Fourier series termwise. – Daniel Fischer Sep 19 '15 at 08:59
This comment and thread and this Bernoulli polynomials link may help too. – Raymond Manzoni Sep 19 '15 at 09:20
Please define c() – Jerry Guern Sep 19 '15 at 10:02
I think $c(x-\frac{1}{2})$ means: "a constant multiple of $x-\frac{1}{2}$." – GEdgar Sep 19 '15 at 13:56
Yes, it is Bernoulli polynomial of degree $s+1$. Thank you for the link. – Willie Wu Sep 20 '15 at 07:10

what is it if the Fourier series is $\sum_{n=1}^\infty \frac{\cos 2n\pi x}{n^{s+1}}$

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