Transfer matrix

The transfer matrix appears in the treatment of refinable functions.

For the mask $h$, which is 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

$T_{h}={\begin{pmatrix}h_{a}&&&&&\\h_{a+2}&h_{a+1}&h_{a}&&&\\h_{a+4}&h_{a+3}&h_{a+2}&h_{a+1}&h_{a}&\\\ddots &\ddots &\ddots &\ddots &\ddots &\ddots \\&h_{b}&h_{b-1}&h_{b-2}&h_{b-3}&h_{b-4}\\&&&h_{b}&h_{b-1}&h_{b-2}\\&&&&&h_{b}\end{pmatrix}}$

Properties

• If you drop the first and the last column and move the odd indexed columns to the left and the even indexed columns to the right, then you obtain a transposed Sylvester_matrix.
• The determinant of a transfer matrix is essentially a resultant.

More precisely:

Let $h_{\mathrm {e} }$ be the even indexed coefficients of $h$ ($(h_{\mathrm {e} })_{k}$=$h_{2\cdot k}$) and let $h_{\mathrm {o} }$ be the odd indexed coefficients of $h$ ($(h_{\mathrm {o} })_{k}$=$h_{2\cdot k+1}$).

Then $\det T_{h}=(-1)^{\lfloor {\frac {b-a+1}{4}}\rfloor }\cdot h_{a}\cdot h_{b}\cdot \mathrm {res} (h_{\mathrm {e} },h_{\mathrm {o} })$, where $\mathrm {res}$ is the resultant.

This connection allows for fast computation using the [[Euclidean algorithm].

References

• Henning Thielemann: Optimally matched wavelets (contains proofs of the above properties)