From big q-Jacobi and Chebyshev polynomials to exponential-reproducing subdivision: new identities

TL;DR

The paper addresses the problem of obtaining explicit closed-form symbols for minimum-support interpolating subdivision schemes that reproduce finite sets of exponential polynomials. It develops new algebraic identities linking Chebyshev polynomials and big $q$-Jacobi polynomials through ${}_3phi_2$ sums, and uses these to construct an explicit $k$-level symbol $m_{2n+2,k}(z)$ for the non-stationary scheme reproducing the space $V_{2n+2, heta}$. Key contributions include a reciprocal identity $(T_n(x))^{-1}=P_n(t; -t^{-1},-t^{-1},-1; t^2)$, a supporting ${}_3phi_2$ summation identity, and a closed-form expression for the scheme symbol that converges to the $(2n+2)$-point Dubuc–Deslauriers symbol as $k o obreak\nobreak fty$, with concrete examples for small $n$. The work yields exact, implementable subdivision tools for exponential-reproducing spaces, with potential impact in geometric modeling and approximation theory, and opens avenues for applying the derived Chebyshev/$q$-Jacobi identities beyond subdivision.

Abstract

In this paper we derive new identities satisfied by Chebyshev polynomials of the first kind and big q-Jacobi polynomials. An immediate benefit of the derived identities is the achievement of closed-form expressions for the Laurent polynomials that identify minimum-support interpolating subdivision schemes reproducing finite sets of integer powers of exponentials.

Figures (4)

• Figure 1: Examples of $\nu$-pointed star-shaped curves with parametric equations $(x(u),y(u))$ (first row) and $(x(u),y(u),z(u))$ (second row) where $x(u)=( 3+\sin(\nu u) ) \cos(u)$, $y(u)=( 3+\sin(\nu u) ) \sin(u)$, $z(u)=-(3+\sin(\nu u) )^2/4$, $u \in [0, 2\pi]$, obtained after 6 subdivision steps of the interpolatory schemes $m_{2n+2,k}(z)$ reproducing $V_{2n+2,\theta}$ with $n=\nu+1$ (first row) while $n=2\nu$ (second row) and $\theta=(2\pi)/(N-1)$, starting from the dashed polygons with vertices ${\bf p}_i^{(0)}=(x_i^{(0)}, y_i^{(0)})$, $i=1,\ldots,N$ (first row) and ${\bf p}_i^{(0)}=(x_i^{(0)}, y_i^{(0)}, z_i^{(0)})$, $i=1,\ldots, N$ (second row) where $x_i^{(0)}= \left( 3+\sin \left(\nu \theta (i-1) \right) \right) \cos \left( \theta (i-1) \right)$, $y_i^{(0)}=\left( 3+\sin \left( \nu \theta (i-1) \right) \right) \sin \left( \theta (i-1) \right)$, $z_i^{(0)}=-\left(3+\sin \left(\nu \theta (i-1) \right) \right)^2/4$.
• Figure 2: Examples of planar Lissajous curves with parametric equations $x(u)= \cos(\nu_2 u)$, $y(u)= \cos(\nu_1 u-(\tau \pi)/\nu_2 )$, with $u \in [0, 2\pi]$, obtained after 6 subdivision steps of the interpolatory schemes $m_{10,k}(z)$ (first two columns) and $m_{12,k}(z)$ (last two columns) reproducing $V_{2n+2,\theta}$ with $\theta=(2\pi)/(N-1)$ and $n=\max\{\nu_1,\nu_2\}$, starting from the dashed polygons with vertices ${\bf p}_i^{(0)}=(x_i^{(0)}, y_i^{(0)})$, $i=1,\ldots, N$ where $x_i^{(0)}= \cos \left(\nu_2 \theta (i-1) \right)$, $y_i^{(0)}= \cos \left(\nu_1 \theta (i-1) -(\tau \pi)/ \nu_2 \right)$.
• Figure 3: Examples of spatial Lissajous curves with parametric equations $x(u)=\cos(\nu_1 u)$, $y(u)=\cos(\nu_2 u)$, $z(u)=\cos(\nu_3 u)$ with $u \in [0,\pi]$, obtained after 6 subdivision steps of the interpolatory schemes $m_{12,k}(z)$ (left), $m_{14,k}(z)$ (center), $m_{16,k}(z)$ (right) reproducing $V_{2n+2,\theta}$ with $\theta=\pi/(N-1)$ and $n=\max\{\nu_1, \nu_2, \nu_3\}$, starting from the dashed polygons with vertices ${\bf p}_i^{(0)}=(x_i^{(0)}, y_i^{(0)}, z_i^{(0)})$, $i=1,\ldots, N$ where $x_i^{(0)}=\cos \left(\nu_1 \theta (i-1) \right)$, $y_i^{(0)}=\cos \left(\nu_2 \theta (i-1) \right)$, $z_i^{(0)}=\cos \left(\nu_3 \theta (i-1) \right)$.
• Figure 4: Examples of spherical Lissajous curves with parametric equations $x(u)=\sin(\nu_2 u) \cos(\nu_1 u - \rho \pi)$, $y(u)=\sin(\nu_2 u) \sin(\nu_1 u - \rho \pi)$, $z(u)=\cos(\nu_2 u)$ with $u \in [0,2\pi]$, obtained after 6 subdivision steps of the interpolatory schemes $m_{12,k}(z)$ (left), $m_{14,k}(z)$ (center), $m_{16,k}(z)$ (right) reproducing $V_{2n+2,\theta}$ with $\theta=(2 \pi)/(N-1)$ and $n=\nu_1+\nu_2$, starting from the dashed polygons with vertices ${\bf p}_i^{(0)}=(x_i^{(0)}, y_i^{(0)}, z_i^{(0)})$, $i=1,\ldots, N$ where $x_i^{(0)}=\sin \left(\nu_2 \theta (i-1) \right) \cos \left(\nu_1 \theta (i-1) - \rho \pi \right)$, $y_i^{(0)}=\sin \left(\nu_2 \theta (i-1) \right) \sin \left(\nu_1 \theta (i-1) - \rho \pi \right)$, $z_i^{(0)}=\cos \left(\nu_2 \theta (i-1) \right)$.

Theorems & Definitions (20)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 3.1
• proof
• Remark 3.2
• Lemma 3.3
• proof
• Proposition 3.4