Scaling function and refinement equation

Summary

The scaling function $\varphi$ of a multiresolution analysis is a function whose integer translates form an orthonormal basis of the base space $V_0$. Since $V_0 \subseteq V_1$, it satisfies a refinement equation $\varphi(x) = \sqrt2 \sum_k h_k \varphi(2x - k)$: it is a combination of half-size copies of itself. The coefficients $h_k$ form the low-pass filter of the discrete wavelet transform, and they also determine the wavelet. Conversely, one constructs wavelets such as the Daubechies wavelets by choosing the coefficients first and solving the refinement equation.