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There is, because of <math>V_0\subset V_1</math>, a finite sequence of coefficients <math>a_k=2 \langle\phi(x),\phi(2x-k)\rangle</math>, for <math>|k|\leq N</math> and <math>a_k=0</math> for <math>|k|>N</math>, such that
:<math>\phi(x)=\sum_{k=-N}^N a_k\phi(2x-k).</math>
Defining another function, known as '''mother wavelet''' or just '''the wavelet'''
:<math>\psi(x):=\sum_{k=-N}^N (-1)^k a_{1-k}\phi(2x-k),</math>
one can see that the space <math>W_0\subset V_1</math>, which is defined as the linear hull of the mother wavelets integer shifts, is the orthogonal complement to <math>V_0</math> inside <math>V_1</math>. Or put differently, <math>V_1</math> is the orthogonal sum of <math>W_0</math> and <math>V_0</math>. By self-similarity, there are scaled versions <math>W_k</math> of <math>W_0</math> and by completeness one has
:<math>L^2(\mathbb R)=\mbox{closure of }\bigoplus_{k\in\Z}W_k,</math>
thus the set
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