On the critical length conjecture for spherical Bessel functions in CAGD

TL;DR

This paper proves a conjecture relating the critical length of certain function spaces generated by x^k sin x and x^k cos x to the zeros of Bessel functions, with implications for Hankel matrix determinants.

Contribution

It confirms the conjecture by Carnicer, Mainar, and Peña, and extends the result to various generalizations involving Bessel functions and Hankel matrices.

Findings

Confirmed the conjecture (D3) about the non-vanishing of a Hankel matrix determinant in a specific interval.

Extended the conjecture to broader classes of functions and matrices.

Provided mathematical proof for the critical length conjecture in CAGD context.

Abstract

A conjecture of J.M. Carnicer, E. Mainar and J.M. Pe\~{n}a 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 $f_{n}\left( x\right) =\sqrt{\frac{\pi}{2}} x^{n+\frac{1}{2}}J_{n+\frac{1}{2}}\left( x\right) .$ In this paper we shall prove (D3) and various generalizations.