On the stabilization of the inversion of some Kontorovich-Lebedev like integral transforms
The authors deal with the Kontorovich-Lebedev transform \[ K_{ix}[f](x)=\text{cosh}(\pi x)\int^\infty_0K_{ix}(y)f(y)dy= g(x)\quad (x\geq 0)\tag{*} \] on the weighted \(L_p\)-space \(L_{\nu,p}(\mathbb{R}_+)\) \((1\leq p\leq\infty,\;\nu\in\mathbb{R})\) normed by \[ \| f\|_{\nu,p}=\left(\int^\infty_0x^{\nu p-1}\bigl| f(x)\bigr|^pdx\right)^{1/p} \] where \(K_{ix}\) is the Macdonald function. By introducing a certain regularization operator they derive a special construction of the inversion of transform \((*)\) that allows to stabilize it in the space \(L_{1+{1\over p},p}(\mathbb{R}_+)\) \((1<p<2)\). Such a construction for the inversion formula concerning the Lebedev-Skalskaya transform [see \textit{N. N. Lebedev} and \textit{I. P. Skalskaya} in: Questions of mathematical physics. Leningrad: Nauka, 68-79 (1976)] \[ \text{Re}[f](x)=\text{cosh}(\pi x)\int^\infty_0\text{Re}K_{{1\over 2}+ix}(y)f(y)dy=g(x)\;(x\geq 0) \] where \(\text{Re} K_{{1\over 2}+ix}(y)\) is the real part of the Macdonald function with the index \({1\over 2}+ix\), is also similarly treated.
scientific article; zbMATH DE number 1211546