A refinement of the two-radius theorem on the Bessel--Kingman hypergroup

Vit. V. Volchkov ; G. V. Krasnoschyokikh

Matematičeskie zametki, Tome 116 (2024) no. 2, pp. 212-228.

Résumé

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.

Détail

Keywords: generalized translation
Mots-clés : convolution equation, Fourier–Bessel transform.

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