= Solution
Write $c_m=\langle f,g_m\rangle$. A <linear N-term approximation> fixes the indices independently of $f$, normally the first $N$ in a prescribed ordering:
$$
f_N^l=\sum_{m=1}^Nc_mg_m.
$$
A <best N-term approximation> chooses the indices using $f$: retain $N$ coefficients of largest absolute value, resolving ties arbitrarily, and set $f_N^n=\sum_{m\in\Lambda_N(f)}c_mg_m$. The <Parseval identity> shows why this choice is optimal:
$$
\boxed{\|f-f_N^n\|^2=\inf_{|\Lambda|\le N}\sum_{m\notin\Lambda}|c_m|^2.}
$$
For any fixed index set, the <orthogonal projection> coefficients minimize the error; optimizing the set then means discarding the smallest squared coefficients. Thus “linear” requires a specified ordering, while “nonlinear” refers to the data-dependent selection.
Back to article page
