Abstract:
We study the behavior at infinity of the solutions of the convolution equations related to the Bessel generalized shift operator. We consider the case where one of the convolutors of the equation is either the characteristic function of the interval or the Dirac measure with support at a given point. Recent results of the authors of the present paper are applied for finding sharp characteristics of the decay rate of nonzero solutions of these equations in terms of the behavior of their integral means. As a corollary, we establish analogs of the well-known theorems by John, Smith, Volchkov, and Thangavelu on injectivity of the spherical mean-value operator on $\mathbb{R}^n$. In addition, in some cases, we strengthen Selmi and Nessibi's theorem on spectral analysis on the Bessel–Kingman hypergroup; we also prove a new uniqueness theorem for the generalized Euler–Poisson–Darboux equation.
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