The one radius theorem for the Bessel convolution operator and its applications

G. V. Krasnoschekikh, Vit. V. Volchkov

Donetsk State University, 24 Universitetskaya St., Donetsk 283001, Russia

Abstract: It is well known that any function $f\in C(\mathbb{R}^n)$ ($n\geq2$) which has zero integrals over all balls and spheres with fixed radius $r$ is identically zero. In this paper, we study a similar phenomenon for ball and spherical means with respect to the $\alpha$-convolution of Bessel. Let $\alpha\in(-1/2,+\infty)$, let $L^{1,\mathrm{loc}}_{\natural,\alpha}(-R,R)$ be the class of even locally summable functions with respect to the measure $d\mu_\alpha(x)=|x|^{2\alpha+1}dx$ on the interval $(-R,R)$, and let $f\overset{\alpha}\star g$ be the Bessel convolution of a function $f\in L^{1,\mathrm{loc}}_{\natural,\alpha}(-R,R)$ and an even distribution $g$ on $\mathbb{R}$ with support in $(-R,R)$. The main result of the article provides a solution to the problem of injectivity of the operator
\begin{equation*} f\rightarrow(f\overset{\alpha}\star\chi_r, f\overset{\alpha}\star\delta_r), f\in L^{1,\mathrm{loc}}_{\natural,\alpha}(-R,R), 0<r<R, \end{equation*}
where $\chi_r$ is the indicator of the segment $[-r,r]$ and $\delta_r$ is an even measure that maps an even continuous function $\varphi$ on $\mathbb{R}$ to the number $\varphi(r)$. Based on the technique associated with classical orthogonal polynomials and recent research by the authors it is shown that for $R\geq2r$, the kernel of the specified operator is zero. For $r<R<2r$, it consist of functions $f\in L^{1,\mathrm{loc}}_{\natural,\alpha}(-R,R)$ that are zero in the interval $(2r-R,R)$ and have a zero integral (with respect to the measure $d\mu_\alpha$) over the interval $(0,2r-R)$. This result allowed us to obtain a new criterion of the closure for the system of generalized Bessel shifts of segment indicators in the space $L^p_{\natural,\alpha}(-R,R)$, $1\leq p<\infty$, as well as a new uniqueness theorem for solutions of the Cauchy problem for the generalized Euler–Poisson–Darboux equation.

Key words: generalized shift, mean periodicity, Gegenbauer polynomials, approximation by shifts, Euler–Poisson–Darboux equation.

UDC: 517.5

MSC: 42A85, 41A30, 33С10, 35Q05

Received: 13.08.2024

Language: English

DOI: 10.46698/e5897-8783-0193-o