On the spectrum of the distributional kernel related to the residue (Q5957255)

Summary: We study the spectrum of the distributional kernel \(K_{\alpha,\beta}(x)\), where \(\alpha\) and \(\beta\) are complex numbers and \(x\) is a point in the space \(\mathbb{R}^n\) of the \(n\)-dimensional Euclidean space. We find that for any nonzero point \(\xi\) that belongs to such a spectrum, there exists the residue of the Fourier transform \((-1)^k\widehat{K_{2k,2k}}(\xi)\), where \(\alpha= \beta= 2k\), \(k\) is a nonnegative integer and \(\xi\in \mathbb{R}^n\).