On the critical length conjecture for spherical Bessel functions in CAGD
Ognyan Kounchev, Hermann Render
TL;DR
This work tackles the CMP-conjecture that the critical length of the translation-invariant space $P_n\odot C_1$ equals the first positive zero $j_{n+\frac{1}{2},1}$ of the spherical Bessel function $J_{n+\frac{1}{2}}$. By introducing a general ODE framework with $f''+p f'+q f=0$ and studying the associated determinants $v(f)$ and $w(f)$, the authors derive positivity and monotonicity properties that govern these Hankel/Wronskian-type objects. They obtain explicit formulas and differential identities, notably $w=(p' f'+q' f)^2 f - A v$ and, when $q'=0$, $w'+\frac{3}{2}(p-\frac{p''}{p'}) w = p' f' V$, with $V$ defined in terms of $p$ and $f$, enabling positivity proofs. Specializing to $f_n(x)=\sqrt{\frac{\pi}{2}}\,x^{n+\frac{1}{2}}J_{n+\frac{1}{2}}(x)$, they prove $w(f_n)>0$ on $(0, j_{n+\frac{1}{2},1})$ for $n=2$ (and previously for $n=1$), thus validating the CMP-conjecture for these cases and providing a robust, general method to verify critical-length statements via positivity of $v$, $w$, and $V$ for a broad class of differential equations.
Abstract
A conjecture of J.M. Carnicer, E. Mainar and J.M. Peña states that the critical length of the space $P_{n}\odot C_{1}$ generated by the functions $x^{k}\sin x$ and $x^{k}\cos x$ for $k=0,...n$ is equal to the first positive zero $j_{n+\frac{1}{2},1}$ of the Bessel function $J_{n+\frac{1}{2}}$ of the first kind. It is known that the conjecture implies the following statement (D3): the determinant of the Hankel matrix \begin{equation} \left( \begin{array} [c]{ccc} f & f^{\prime} & f^{\prime\prime}\\ f^{\prime} & f^{\prime\prime} & f^{\left( 3\right) }\\ f^{\prime\prime} & f^{\prime\prime\prime} & f^{\left( 4\right) } \end{array} \right) \label{eqabstract} \end{equation} does not have a zero in the interval $(0,j_{n+\frac{1}{2},1})$ whenever $f=f_{n}$ is given by $f_{n}\left( x\right) =\sqrt{\fracπ{2}} x^{n+\frac{1}{2}}J_{n+\frac{1}{2}}\left( x\right) .$ In this paper we shall prove (D3) and various generalizations.