Multiresolution Analysis
David F. Walnut · Applied and numerical harmonic analysis · 2004
In Section 5.5, we saw that if h(x) = χ [0,1/2)(x) − { h j,k ( x ) } j,k∈z = { 2 j/2 h( 2 j x−k ) } j,k∈z = { D 2 j T k h( x ) } j,k∈z ]] <![CDATA[$${\left\{ {{h_{j,k}}\left( x \right)} \right\}_{j,k \in z}} = {\left\{ {{2^{j/2}}h\left( {{2^j}x - k} \right)} \right\}_{j,k \in z}} = {\left\{ {{D_{{2^j}}}{T_k}h\left( x \right)} \right\}_{j,k \in z}}$$ forms an orthonormal basis on R.