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.

, which is a vector with component indexes from

, the transfer matrix of

, we call it

here, is defined as More verbosely The effect of

can be expressed in terms of the downsampling operator "

": More precisely: Let

be the even-indexed coefficients of

be the odd-indexed coefficients of

det

{\displaystyle \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} })}

{\displaystyle \mathrm {res} }

is the resultant.

det

= det

⋅ det

{\displaystyle \det T_{g*h}=\det T_{g}\cdot \det T_{h}\cdot \mathrm {res} (g_{-},h)}

⋅ x = λ ⋅ x

is an eigenvector of

with respect to the same eigenvalue, i.e. Let

be the periodization of

with respect to period

is a circular filter, which means that the component indexes are residue classes with respect to the modulus

Then with the upsampling operator

it holds

{\displaystyle \mathrm {tr} (T_{h}^{n})=\left(C_{k}h*(C_{k}h\uparrow 2)*(C_{k}h\uparrow 2^{2})*\cdots *(C_{k}h\uparrow 2^{n-1})\right)_{[0]_{2^{n}-1}}}

ϱ (

is the size of the filter and if all eigenvalues are real, it is also true that

ϱ (