On the critical length conjecture for spherical Bessel functions in CAGD

2025/05/15 by Ognyan Kounchev, Kounchev, Ognyan, Hermann Render +1
Mathematics · #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2505.09964

Abstract

A conjecture of J.M. Carnicer, E. Mainar and J.M. Peña states that the critical length of the space P_n\odot C₁ generated by the functions x^ksin x and x^kcos x for k=0,...n is equal to the first positive zero j_n+(1)/(2),1 of the Bessel function J_n+(1)/(2) of the first kind. It is known that the conjecture implies the following statement (D3): the determinant of the Hankel matrix ( f · amp; f^′ · amp; f^′′
f^′ · amp; f^′′ · amp; f^{( 3)}
f^′′ · amp; f^′′′ · amp; f^{( 4)} ) does not have a zero in the interval (0,j_n+(1)/(2),1) whenever f=f_n is given by f_n( x) =√\fracπ2 x^n+(1)/(2)J_n+(1)/(2)( x) . In this paper we shall prove (D3) and various generalizations.

Citations

• Critical length: an alternative approach

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