On the critical length conjecture for spherical Bessel functions in CAGD
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910945087848448 |
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| author | Kounchev, Ognyan Render, Hermann |
| author_facet | Kounchev, Ognyan Render, Hermann |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09964 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the critical length conjecture for spherical Bessel functions in CAGD Kounchev, Ognyan Render, Hermann Classical Analysis and ODEs Numerical Analysis 41A05, 41A10, 42A10, 65D05, 65D17 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. |
| title | On the critical length conjecture for spherical Bessel functions in CAGD |
| topic | Classical Analysis and ODEs Numerical Analysis 41A05, 41A10, 42A10, 65D05, 65D17 |
| url | https://arxiv.org/abs/2505.09964 |