Ball averages relative to the Bessel convolution and their application

Let $\alpha\in(-1/2,+\infty)$ and $\chi_r$ be the indicator function of the segment $[-r,r]$. New two-radii theorems have been obtained for the Bessel convolution operator $f\rightarrow f\overset{\alpha}\star\chi_r$ related to quasi-analytic classes of functions. A local analogue of the two-radii theorem has also been established for functions $f$ that satisfy the system of convolutional inequalities $f\overset{\alpha}\star\chi_{r_1}\geq0$, $f\overset{\alpha}\star\chi_{r_2}\leq0$. Applications of these results to the uniqueness theorems for solutions of the Cauchy problem for the generalized Euler-Poisson-Darboux equation and closure theorems for generalized shifts are presented.

УДК 517.5

Keywords: generalized shift, Fourier-Bessel transform, Euler-Poisson-Darboux equation, two-radii theorems.