Geodesic
A refinement of the two-radius theorem on the Bessel--Kingman hypergroup
Matematičeskie zametki, Tome 116 (2024) no. 2, pp. 212-228

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In the present paper, we study an equation of the form $$ \int_{0}^{r}T^\alpha_yf(x)x^{2\alpha+1}\,dx=0, \qquad |y| R-r, \quad 0, $$ where $\alpha>-1/2$, $T^\alpha_y$ is the generalized Bessel translation operator, and $f$ is an even function locally integrable with respect to the measure $|x|^{2\alpha+1}\,dx$ on the interval $(-R,R)$. A description of the solutions of this equation in the form of series in special functions is obtained. Based on this result, we completely study the existence of a nonzero solution of a system of two such equations.
Keywords: generalized translation
Mots-clés : convolution equation, Fourier–Bessel transform. 
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     author = {Vit. V. Volchkov and G. V. Krasnoschyokikh},
     title = {A refinement of the two-radius theorem on the {Bessel--Kingman} hypergroup},
     journal = {Matemati\v{c}eskie zametki},
     pages = {212--228},
     publisher = {mathdoc},
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     year = {2024},
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Vit. V. Volchkov; G. V. Krasnoschyokikh. A refinement of the two-radius theorem on the Bessel--Kingman hypergroup. Matematičeskie zametki, Tome 116 (2024) no. 2, pp. 212-228. http://geodesic.mathdoc.fr/item/MZM_2024_116_2_a3/