More precisely:

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

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

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

${\displaystyle g_{-}}$

${\displaystyle (g_{-})_{k}=(-1)^{k}\cdot g_{k}}$

${\displaystyle x}$

${\displaystyle T_{h}}$

${\displaystyle \lambda }$

${\displaystyle T_{h}\cdot x=\lambda \cdot x}$,

then ${\displaystyle x*(1,-1)}$ is an eigenvector of ${\displaystyle T_{h*(1,1)}}$ with respect to the same eigenvalue, i.e.

${\displaystyle T_{h*(1,1)}\cdot (x*(1,-1))=\lambda \cdot (x*(1,-1))}$

${\displaystyle \lambda _{a},\dots ,\lambda _{b}}$

${\displaystyle T_{h}}$

${\displaystyle \lambda _{a}+\dots +\lambda _{b}=\mathrm {tr} ~T_{h}}$

${\displaystyle \lambda _{a}^{n}+\dots +\lambda _{b}^{n}=\mathrm {tr} (T_{h}^{n})}$

${\displaystyle T_{h}}$

${\displaystyle n}$

Let ${\displaystyle C_{k}h}$ be the periodization of ${\displaystyle h}$ with respect to period ${\displaystyle 2^{k}-1}$. That is ${\displaystyle C_{k}h}$ is a circular filter, which means that the component indexes are residue classes with respect to the modulus ${\displaystyle 2^{k}-1}$. Then with the upsampling operator ${\displaystyle \uparrow }$ it holds

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

${\displaystyle n-2}$

${\displaystyle 2\cdot \log _{2}n}$

${\displaystyle \varrho (T_{h})}$

${\displaystyle \varrho (T_{h})\geq {\frac {a}{\sqrt {\#h}}}\geq {\frac {1}{\sqrt {3\cdot \#h}}}}$

where ${\displaystyle \#h}$ is the size of the filter and if all eigenvalues are real, it is also true that

${\displaystyle \varrho (T_{h})\leq a}$,

${\displaystyle a=\Vert C_{2}h\Vert _{2}}$