On the critical length conjecture for spherical Bessel functions in CAGD

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 $k=0,\mathrm{.}\mathrm{.}\mathrm{.}n$ 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 $f_n\left(x\right)=\sqrt{\frac{\pi }{2}}{x}^{n+\frac{1}{2}}{J}_{n+\frac{1}{2}}\left(x\right)\mathrm{.}$ In this paper we shall prove (D3) and various generalizations.