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 \(f\rightarrow(f\overset{\alpha}\star\chi_r, f\overset{\alpha}\star\delta_r), \quad f\in L^{1,\mathrm{loc}}_{\natural,\alpha}(-R,R), \quad 0 < r < R,\) 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.
Keywords: generalized shift, mean periodicity, Gegenbauer polynomials, approximation by shifts, Euler-Poisson-Darboux equation
For citation: Krasnoschekikh, G. V. and Volchkov, Vit. V. The One Radius Theorem for the Bessel Convolution Operator and its Applications, Vladikavkaz Math. J., 2025, vol. 27, no. 2, pp. 72-83. DOI 10.46698/e5897-8783-0193-o
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