Transfer matrix

In applied mathematics, the transfer matrix is a formulation in terms of a block-Toeplitz matrix of the two-scale equation, which characterizes refinable functions. Refinable functions play an important role in wavelet theory and finite element theory.

For the mask $$h$$, which is a vector with component indexes from $$a$$ to $$b$$, the transfer matrix of $$h$$, we call it $$T_h$$ here, is defined as

(T_h)_{j,k} = h_{2\cdot j-k}. $$ More verbosely

T_h = \begin{pmatrix} h_{a } &         &         &         &         &   \\ h_{a+2} & h_{a+1} & h_{a } &         &         &   \\ h_{a+4} & h_{a+3} & h_{a+2} & h_{a+1} & h_{a } &   \\ \ddots & \ddots  & \ddots  & \ddots  & \ddots  & \ddots \\ & h_{b } & h_{b-1} & h_{b-2} & h_{b-3} & h_{b-4} \\ &        &         & h_{b  } & h_{b-1} & h_{b-2} \\ &        &         &         &         & h_{b  } \end{pmatrix}. $$ The effect of $$T_h$$ can be expressed in terms of the downsampling operator "$$\downarrow$$":
 * $$T_h\cdot x = (h*x)\downarrow 2.$$