The solutions of the $n$-dimensional Bessel diamond operator and the Fourier--Bessel transform of their convolution
Huseyin Yildirim, M Zeki Sarikaya, Sermin Ozturk
TL;DR
The paper studies the n-dimensional Bessel diamond operator $\Diamond_{B}^{k}$, defined from Bessel-type radial operators, and investigates its elementary solutions as well as the Fourier–Bessel transform of these solutions and their convolution. It factors $\Diamond_{B}^{k}$ as $\Diamond_{B}^{k}=\Box_B^k\Delta_B^k$, introduces fundamental solutions $E(x)$ for $\Delta_B$, $S_{2k}(x)$ for $\Delta_B^k$, and $R_{2k}(x)$ for $\Box_B^k$, and develops tools including $B$-convolution, the generalized shift $T^y$, and the Fourier–Bessel transform. The main result is that the unique elementary solution of $\Diamond_{B}^{k}$ is $u(x)=(-1)^k\,S_{2k}(x)*R_{2k}(x)$, with $E$ and $S_{2k}$ providing fundamental solutions for the related operators. The work also derives the Fourier–Bessel transform of the elementary solution and of their convolution, illuminating the transform properties and kernels involved.
Abstract
In this article, the operator $\Diamond_{B}^{k}$ is introduced and named as the Bessel diamond operator iterated $k$ times and is defined by $ \Diamond_{B}^{k} = [ (B_{x_{1}} + B_{x_{2}} + ... + B_{x_{p}})^{2} - (B_{x_{p + 1}} + ... + B_{x_{p + q}})^{2} ]^{k}$, where $ p + q = n, B_{x_{i}} = \frac{\partial^{2}}{\partial x_{i}^{2}} + \frac{2v_{i}}{x_{i}} \frac{\partial}{\partial x_{i}}, $ where $2v_{i} = 2\alpha_{i} + 1$, $ \alpha_{i} > - {1/2} $ [8], $x_{i} > 0$, $i = 1, 2, ..., n, k$ is a non-negative integer and $n$ is the dimension of $\mathbb{R}_{n}^{+}$. In this work we study the elementary solution of the Bessel diamond operator and the elementary solution of the operator $\Diamond_{B}^{k}$ is called the Bessel diamond kernel of Riesz. Then, we study the Fourier--Bessel transform of the elementary solution and also the Fourier--Bessel transform of their convolution.