# Is it possible to visualize the graph of a cosine or sine fourier series of an arbitrary function without calculating the coefficients?

Given an arbitrary function (usually a combination of even and odd functions), is it possible to visualize how the sine or cosine fourier series will appear without calculating the coefficients and numerically producing a graph?

• I should note that, for better or for worse, my idea was to assume that the cosine series was simply the even part of the function and ditto for the sine series (except odd).
– xaav
Oct 10, 2016 at 12:20
• Do you mean you'd like some fast way to (approximately) plot a Fourier polynomial for a given $f$, i.e., to send $f$ through a low-pass filter, without calculating all the necessary coefficients? Oct 10, 2016 at 13:05
• The Fourier series is composed of a sine series and a cosine series. What I want is a fast way to plot either only the the sum of the sines or only the sum of the cosines (my choice).
– xaav
Oct 10, 2016 at 14:01

As you note in the comments, if your Fourier series is defined on a symmetric interval $[-\ell, \ell]$, then the cosine series of $f$ is the even part of $f$ and the sine series is the odd part of $f$. These parts are given by simple algebraic formulas: $$f_{\text{even}}(x) = \tfrac{1}{2}\bigl(f(x) + f(-x)\bigr),\qquad f_{\text{odd}}(x) = \tfrac{1}{2}\bigl(f(x) - f(-x)\bigr).$$