N-term approximation ($f_N$)

An N-term approximation in an orthonormal basis uses at most $N$ basis functions. For a chosen index set $\Lambda$, its optimal coefficients are the inner products with those functions, and its squared error is the sum of the omitted squared coefficients by the Parseval identity.

The best N-term approximation in an orthonormal basis retains $N$ coefficients of greatest absolute value. It minimizes the squared Hilbert space error over all index sets of size at most $N$. The selection makes the approximation nonlinear in the data, even though coefficient extraction is linear.

A linear N-term approximation fixes its index set independently of the approximated function. Wavelet approximation normally orders by increasing resolution, whereas Fourier series approximation normally keeps the lowest frequencies. An arbitrary reordering can change or destroy a claimed convergence rate.