Consider the Fourier series of f, defined using the interval −1 ≤ x ≤ 1.
Plot f together with its Fourier series (the plot should show several periods of the Fourier series).
(i) $$f(x) = 1$$
(ii) $$f(x) =x^2$$
What does a plot of these Fourier series look like? Thank you so much for any help or hints!
The Fourier series of a constant is $1$, and the Fourier transform of $t^2$ is $-2 e^{-i t}-2 e^{i t}+\frac{1}{2} e^{-2 i t}+\frac{1}{2} e^{2 i t}-\frac{2}{9} e^{-3 i t}-\frac{2}{9} e^{3 i t}+\frac{1}{8} e^{-4 i t}+\frac{1}{8} e^{4 i t}-\frac{2}{25} e^{-5 i t}-\frac{2}{25} e^{5 i t}+\frac{\pi ^2}{3}$.