= Wavelet approximation complexity
{title2=$V_J=V_{J_0}\oplus\bigoplus_{j=J_0}^{J-1}W_j$}
An orthonormal <multiresolution analysis> separates coarse <scaling functions> from progressively finer <wavelet> details. On a bounded interval, the approximation space at level $J$ has dimension of order $2^J$. Higher resolution reduces projection bias but estimates more noisy coefficients. In an orthonormal Gaussian sequence model with coefficient noise variance $\sigma^2/n$, risk is the omitted squared coefficient sum plus $\sigma^2d_J/n$. Selecting or shrinking individual detail coefficients yields spatially adaptive complexity; a data-selected subset need not have effective degrees of freedom equal to its raw selected count.
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