Measures that can be used to truncate linear series of functions
We are in the realm of non-linear data approximation. Among the standard litterature, there are Nonlinear approximation, 1998 (R. de Vore), or the more visual On Fourier and Wavelets: Representation, Approximation and Compression, 2006, Vetterli.

Best $N$-term approximation, with respect to some norm $\|f-f_N \|_p$, is quite involved (generically, NP-hard). A basic (suboptimal) approach keeps the $N$ biggest terms (esp. with an orthogonal basis)), which (since $|c_k|\ge|c_{k+1}|$) amounts to finding a proper threshold $\gamma$, and perfom hard-thresholding, up to $|c_N|\ge \gamma &gt;|c_{N+1}|$. There are many strategies to pick an optimal threshold, with additional assumptions on the data, on the noise. 

Since you are interested in the Akaike Information Criterion (AIC), you can check A Crossvalidatory AIC for Hard Wavelet Thresholding in Spatially Adaptive Function Estimation, 1998, that compares with other standard wavelet shrinkages. 

In general, you have to balance a number of terms (on even bits) with the accuracy. The $\ell_0$ constraint can be converted into other norm penalty. If you want to compress, denoise, you can use other sparsity measures like entropy-based method. Further, you have all sort of matching techniques. You can have a quick overview, for images, in Chapter 4: Redundancy and adaptivity.