On the zeros of mean-periodic functions with respect to the Bessel convolution

Vit. V. Volchkov   ; G. V. Krasnoschekikh

Čebyševskij sbornik, Tome 25 (2024) no. 5, pp. 57-73

Résumé

In paper, we study uniqueness sets for solutions to the Bessel convolution equation $f\overset{\alpha}\star g=0$, $\alpha\in(-1/2,+\infty)$. It is shown, in particular, that if $g=\chi_r$ is an indicator function of the segment $[-r,r]$, and an even function $f\in C(\mathbb{R})$ satisfies the equation $f\overset{\alpha}\star\chi_r=0$ and is zero on $(r-\varepsilon,r)$ or $(r,r+\varepsilon)$ for some $\varepsilon>0$, then $f=0$ on $(r-\varepsilon,r+\varepsilon)$. In this case, the interval of zeros $(r-\varepsilon,r+\varepsilon)$, cannot generally be extended. It has been established that a similar phenomenon occurs for solutions of the equation $f\overset{\alpha}\star\delta_r=0$, where $\delta_r$ is an even measure that maps an even continuous function $\varphi$ on $\mathbb{R}$ to a number $\varphi(r)$. Applications of these results to uniqueness theorems for convergent sequences of linear combinations of Bessel functions, the zero set theorem for solutions of the Cauchy problem of the generalized Euler-Poisson-Darboux equation and the closure theorem of generalized shifts are found.

Détail

Keywords: generalized convolution, spherical transformation, uniqueness sets, shift closure theorems, lacunar series.