 Solution

Write $c_m=⟨f,g_m⟩$. A linear N-term approximation fixes the indices independently of $f$, normally the first $N$ in a prescribed ordering:A best N-term approximation chooses the indices using $f$: retain $N$ coefficients of largest absolute value, resolving ties arbitrarily, and set $f_N^n={\sum }_{m\in {\Lambda }_N(f)}{c}_m{g}_m$. The Parseval identity shows why this choice is optimal: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.