Wavelet approximation complexity ($V_J=V_{J_0}\oplus {⨁}_{j=J_0}^{J-1}{W}_j$)

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 ${\sigma }^2{d}_J/n$. 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.