Joint spectral radius and ternary hermite subdivision

Abstract

In this paper we construct a family of ternary interpolatory Hermite subdivision schemes of order 1 with small support and ${\mathscr{H}}\mathcal {C}^{2}$-smoothness. Indeed, leaving the binary domain, it is possible to derive interpolatory Hermite subdivision schemes with higher regularity than the existing binary examples. The family of schemes we construct is a two-parameter family whose ${\mathscr{H}}\mathcal {C}^{2}$-smoothness is guaranteed whenever the parameters are chosen from a certain polygonal region. The construction of this new family is inspired by the geometric insight into the ternary interpolatory scalar three-point subdivision scheme by Hassan and Dodgson. The smoothness of our new family of Hermite schemes is proven by means of joint spectral radius techniques.

Appendix

For the interested reader, we provide the details of our MATLAB routines via MATLAB exchange, see t-toolboxes [30]. For completeness, we provide here the transition matrices $\left \{T_{\varepsilon }^{{\boldsymbol {A}}}|_{V_{2}^{{\boldsymbol {A}}}} : \varepsilon \in \{0,1,2\}\right \}$

$$ \begin{array}{c} \frac{1}{729} \left( \begin{array}{ccccccc} 64\mu + 243 & 96\mu + 108 & 180-96\mu & -112\mu-144 & -80\mu-72 \\ 64\lambda-128\mu-480 & -96\lambda-192\mu-57 & 192\mu-360 & 224\mu-224\lambda + 462 & 96\lambda + 160\mu-6 \\ 128\lambda-60 & 192\lambda-30 & -192\lambda-33 & 39-224\lambda & 21-160\lambda \\ 16\mu + 99 & 72\mu + 81 & 90-48\mu & 0 & -64\mu-63 \\ -112\mu-369 & 9-120\mu & 144\mu-270 & 224\mu + 423 & 96\mu-54 \end{array}\right),\\ \frac{1}{729} \left( \begin{array}{ccccccc} -96\mu-135 & -48\mu-54 & 96\mu + 108 & 0 & -64\mu-90 \\ 192\mu-192\lambda + 576 & 192\lambda + 96\mu + 222 & -96\lambda-192\mu-57 & 32\lambda & 128\mu-416\lambda + 240 \\ 36-192\lambda & 15-96\lambda & 192\lambda-30 & 0 & 24-128\lambda \\ 0 & -72\mu-81 & 72\mu + 81 & -8\mu-18 & 72\mu + 81 \\ 192\mu + 540 & 24\mu + 171 & 9-120\mu & -8\mu-9 & 200\mu + 252 \end{array}\right),\\ \frac{1}{729} \left( \begin{array}{ccccccc} 32\mu + 45 & 0 & -48\mu-54 & 0 & 0 \\ 128\lambda-64\mu-132 & -192\lambda-33 & 192\lambda + 96\mu + 222 & 32\lambda-9 & 224\lambda + 33 \\ 64\lambda-12 & 0 & 15-96\lambda & 0 & 0 \\ -16\mu-99 & 48\mu-90 & -72\mu-81 & -8\mu-9 & 72-56\mu \\ -80\mu-171 & 48\mu-90 & 24\mu + 171 & -8\mu-18 & 81-56\mu \end{array}\right). \end{array} $$

(1)

Similarly we obtain $\left \{T_{\varepsilon }^{{\boldsymbol {B}}}|_{V_{0}^{{\boldsymbol {B}}}} : \varepsilon \in \{0,1,2\}\right \}$ given by (to be scaled by 1/243)

$$ \begin{array}{c} \left( \begin{array}{ccccccc} 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 423 - 32\mu & 96\mu - 180 & -27 & - 96\mu - 108 & 48\mu + 54 & 16\mu + 9 & 96\mu + 135 \\ 64\lambda + 32\mu + 138 & - 192\lambda - 33 & - 144\mu - 195 & 192\lambda + 48\mu + 24 & - 96\lambda - 96\mu - 93 & - 32\lambda - 64\mu - 87 & 176\mu - 192\lambda + 270 \\ 12 - 64\lambda & 0 & 288\lambda - 51 & 15 - 96\lambda & 192\lambda - 30 & 128\lambda - 24 & 63 - 352\lambda \\ 64\mu + 72 & 0 & - 288\mu - 792 & 96\mu + 252 & - 192\mu - 72 & - 128\mu - 459 & 352\mu + 882 \\ 450 & 96\mu - 180 & - 144\mu - 630 & 90 - 48\mu & 90 - 48\mu & - 48\mu - 360 & 272\mu + 783 \end{array}\right),\\ \left( \begin{array}{ccccccc} - 16\mu - 9 & 48\mu + 54 & 0 & 0 & 0 & 0 & 0 \\ 32\lambda - 16\mu - 12 & 48\mu - 96\lambda + 69 & 0 & 0 & 0 & 0 & 0 \\ 112\mu & 48\mu + 342 & 423 - 32\mu & 96\mu - 180 & - 96\mu - 108 & 16\mu + 324 & - 32\mu - 360 \\ - 320\lambda - 32\mu - 90 & 192\lambda + 96\mu + 222 & 64\lambda + 32\mu + 138 & - 192\lambda - 33 & 192\lambda + 48\mu + 24 & 32\mu - 32\lambda + 117 & 64\lambda - 32\mu - 126 \\ 0 & 0 & 12 - 64\lambda & 0 & 15 - 96\lambda & 12 - 64\lambda & 64\lambda - 12 \\ 32\mu + 180 & - 96\mu - 252 & 64\mu + 72 & 0 & 96\mu + 252 & 64\mu + 81 & - 64\mu - 72 \\ 144\mu + 180 & 90 - 48\mu & 450 & 96\mu - 180 & 90 - 48\mu & 48\mu + 360 & - 64\mu - 387 \end{array}\right),\\ \left( \begin{array}{ccccccc} -27 & - 96\mu - 108 & - 16\mu - 9 & 48\mu + 54 & 0 & - 48\mu - 54 & 48\mu + 54 \\ -33 & 192\lambda - 96\mu - 138 & 32\lambda - 16\mu - 12 & 48\mu - 96\lambda + 69 & 0 & 96\lambda - 48\mu - 69 & 48\mu - 96\lambda + 69 \\ - 144\mu - 1098 & 18 - 240\mu & 112\mu & 48\mu + 342 & 96\mu - 180 & - 96\mu - 261 & 80\mu + 243 \\ 288\lambda - 519 & - 96\lambda - 192\mu - 57 & - 320\lambda - 32\mu - 90 & 192\lambda + 96\mu + 222 & - 192\lambda - 33 & - 96\lambda - 96\mu - 210 & 128\lambda + 96\mu + 210 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 468 & 192\mu + 72 & 32\mu + 180 & - 96\mu - 252 & 0 & 96\mu + 261 & - 96\mu - 270 \\ - 144\mu - 630 & 90 - 48\mu & 144\mu + 180 & 90 - 48\mu & 96\mu - 180 & 0 & - 16\mu - 27 \end{array}\right). \end{array} $$

(2)

The mask B derived using the Taylor operator approach is given by (to be scaled by 1/243)

$$ \begin{array}{ll} {\boldsymbol{B}}(-5) = \left( \begin{array}{cc} - 16\mu - 9 & 32\lambda - 16\mu - 12 \\ 48\mu + 54 & 48\mu - 96\lambda + 69 \end{array}\right), & {\boldsymbol{B}}(-4) = \left( \begin{array}{cc} - 32\mu - 45& 64\lambda - 32\mu - 57 \\ 0& 0 \end{array}\right),\\ {\boldsymbol{B}}(-3) = \left( \begin{array}{cc} 342 - 80\mu & 160\lambda - 80\mu + 414 \\ 48\mu - 234 & 48\mu - 96\lambda - 282 \end{array}\right), & {\boldsymbol{B}}(-2) = \left( \begin{array}{cc} 16\mu + 747 & 819 - 16\mu - 128\lambda \\ 48\mu - 234 & 192\lambda + 144\mu - 138 \end{array}\right), \\ {\boldsymbol{B}}(-1) = \left( \begin{array}{cc} 108 - 32\mu & - 128\lambda - 96\mu - 327 \\ 0 & 243 \end{array}\right), & {\boldsymbol{B}}(0) = \left( \begin{array}{cc} - 80\mu - 99& - 320\lambda - 240\mu - 387\\ 48\mu + 54& 192\lambda + 144\mu + 150 \end{array}\right), \\ {\boldsymbol{B}}(1) = \left( \begin{array}{cc} 32\mu + 45 & 160\lambda + 96\mu + 261 \\ 0 & - 96\lambda - 48 \end{array}\right), & {\boldsymbol{B}}(2) = \left( \begin{array}{cc} 0 & 64\lambda + 24 \\ 0 & 0 \end{array}\right), \\ {\boldsymbol{B}}(3) = \left( \begin{array}{cc} 0 & 160\lambda - 27 \\ 0 & 15 - 96\lambda \end{array}\right), &{\boldsymbol{B}}(4) = \left( \begin{array}{cc} 0 & 12 - 64\lambda\\ 0 & 0 \end{array}\right). \end{array} $$

(3)

Last, we provide the computation of the entries (see (??)) of $\tilde {\boldsymbol {B}}^{*}_{+}(z)= \left (\begin {array}{rrr}\tilde b_{00}(z)& \tilde b_{01}(z)& \tilde b_{02}(z)\\ \tilde b_{10}(z)& \tilde b_{11}(z)& \tilde b_{12}(z)\\ \tilde b_{20}(z)& \tilde b_{21}(z)& \tilde b_{22}(z)\\ \end {array} \right ):$

$$ \begin{array}{ll} \tilde b_{00}(z)&=\frac{80\mu {z^{6}}+99{z^{6}}-32\mu {z^{5}}-45{z^{5}}+16\mu {z^{4}}-126{z^{4}}+112\mu {z^{3}}+279{z^{3}}-32\mu {z^{2}}+108{z^{2}}+16\mu z+9z+32\mu+45}{81{z^{2}}}\\ \tilde b_{01}(z)&= -\frac{320\mu {z^{6}}+640\lambda {z^{6}}+315{z^{6}}-128\mu {z^{5}}-256\lambda {z^{5}}-51{z^{5}}+64\mu {z^{4}}+128\lambda {z^{4}}-273{z^{4}}}{324{z^{2}}}\\ &\quad -\frac{128\mu {z^{3}}-384l {z^{3}}+483{z^{3}}+256\lambda {z^{2}}+177{z^{2}}-128\lambda z+39z-256\lambda+48}{324{z^{2}}}\\ \tilde b_{02}(z)&= \frac{z \left( 160\mu {z^{3}}+640\lambda {z^{3}}+117{z^{3}}-64\mu {z^{2}}-256\lambda {z^{2}}+39{z^{2}}+32\mu z+128\lambda z-21z+64\mu+256\lambda+42\right) }{324}\\ \tilde b_{10}(z)&=\frac{2\left( z-1\right) \left( z+1\right) \left( 8\mu {z^{4}}+9{z^{4}}-8\mu {z^{3}}-9{z^{3}}+16\mu {z^{2}}-30{z^{2}}-8\mu z-9z+8\mu+9\right) }{27{z^{2}}} \\ \tilde b_{11}(z)&= -\frac{\left( z-1\right) \left( 32\mu {z^{5}}+64\lambda {z^{5}}+35{z^{5}}+36{z^{4}}+32\mu {z^{3}}+64\lambda {z^{3}}-43{z^{3}}-64\lambda {z^{2}}-68{z^{2}}-9z-64\lambda+10\right) }{54{z^{2}}} \\ \tilde b_{12}(z)&= \frac{z \left( 16\mu {z^{3}}+64\lambda {z^{3}}+17{z^{3}}-16\mu {z^{2}}-64\lambda {z^{2}}+19{z^{2}}+16\mu z+64\lambda z-z-16\mu-64\lambda-8\right) }{54}\\ \tilde b_{20}(z)&=0\qquad \tilde b_{21}(z)=-\frac{\left( z-1\right) \left( z+1\right) {{\left( {z^{2}}+z+1\right) }^{2}}}{6{z^{2}}}, \quad \tilde b_{22}(z)= \frac{z \left( z+1\right) \left( {z^{2}}+z+1\right) }{6}. \end{array} $$

(4)