Fourier partial sum
= Fourier partial sum
{c}
{title2=$s_n(f)$}
A <Fourier partial sum> retains the modes whose <integer> frequency multipliers satisfy $|k|\leq n$ in a <complex Fourier series>:
$$
S_n f(x)=\sum_{k=-n}^{n}\widehat f_k e^{ikx}.
$$
For a real <function> of period $2\pi$, this is equivalently $a_0/2+\sum_{k=1}^{n}[a_k\cos(kx)+b_k\sin(kx)]$. It is a <trigonometric polynomial>, even when the original <function> is discontinuous. For a <square wave>, retaining $M$ nonzero odd sine modes gives $S_{2M-1}f$; all even-frequency coefficients are zero. The <Gibbs phenomenon> illustrates why more retained modes need not reduce the maximum error.