The ultrahyperbolic Bessel operator: An inversion theorem
Let \(x, t\in \mathbb{R}^n\), \(P(t)= t^2_1+\cdots+ t^2_p- t^2_{p+ 1}-\cdots- t^2_{p+ q}\), \(p+ q= n\). The ultrahyperbolic Bessel operator of order \(\alpha> 0\) is defined by \(B^\alpha\varphi= G_\alpha(P+ i0)* \varphi\), where \[ G_\alpha(r)= {2^{1- (\alpha+ n)/2} |x|^{(\alpha- n)/2}\over \pi^{n/2} \Gamma(\alpha/2)} K_{(n- \alpha)/2}(r), \] \(K_{(n- \alpha)/2}(r)\) being the modified Bessel functions of the third kind. In order to invert \(B^\alpha\) the author introduces the truncated causal generalized Bessel derivative \[ D^\alpha_\varepsilon f= {1\over d_{n,\ell}(\alpha)} \int_{|t|>\varepsilon} {(\Delta^\ell_t f)(x, \lambda_\alpha(P+ i0))\over (P+ i0)^{(n+ \alpha)/2}} dt,\quad \varepsilon> 0, \] with the weighted finite difference \[ (\Delta^\ell_t f)(x, \lambda_\alpha)= \sum^\ell_{k= 0} (\begin{smallmatrix} \ell\\ k\end{smallmatrix}) (- 1)^k \lambda_\alpha(kt) f(x- kt). \] The constant \(d_{n,\ell}(\alpha)\) and the weight function \(\lambda_\alpha\) are determined in the paper. Theorem. Let \(f\) and \(\varphi\) be tempered distributions. The following two statements are equivalent: (1) \(f= B^\alpha\varphi\) and (2) \(\varphi= \lim_{\varepsilon\to 0} D^\alpha_\varepsilon f\).
modified Bessel function of the third kind
truncated causal generalized Bessel derivative
scientific article; zbMATH DE number 810955