On the spectrum of the distributional kernel related to the residue

Abstract

We study the spectrum of the distributional kernel Kα,β(x), where α and β are complex numbers and x is a point in the space ℝn of the n-dimensional Euclidean space. We found that for any nonzero point ξ that belongs to such a spectrum, there exists the residue of the Fourier transform (−1)kK2k,2k(ξ)ˆ, where α=β=2k, k is a nonnegative integer and ξ∈ℝn.