Operators with Complex Gaussian Kernels: Asymptotic Behaviours
Abstract
\begin{abstract}
In this paper we derive Abelian theorems for the operators with complex Gaussian kernels. Specifically, we establish some results in which known the behaviour of the function and its domain variable approaches to $-\infty$ or $+\infty$ is used to infer the asymptotic behaviour of the transform as its domain variable approaches to $+\infty$ or $-\infty$. For this purpose we use a formula concerning the computation of potential functions by means of these operators with complex Gaussian kernels. This formula allows us to analyse the asymptotic behaviour of these operators in both cases: when the variable approaches to $+\infty$ or $-\infty$. Our results include systematically the noncentered and centered cases of these operators. Here we analyse the Gauss-Weierstrass semigroup on $\mathbb{R}$ as a particular case. We also point out Abelian theorems for other kinds of operators which have been studied in several papers.
In this paper we derive Abelian theorems for the operators with complex Gaussian kernels. Specifically, we establish some results in which known the behaviour of the function and its domain variable approaches to $-\infty$ or $+\infty$ is used to infer the asymptotic behaviour of the transform as its domain variable approaches to $+\infty$ or $-\infty$. For this purpose we use a formula concerning the computation of potential functions by means of these operators with complex Gaussian kernels. This formula allows us to analyse the asymptotic behaviour of these operators in both cases: when the variable approaches to $+\infty$ or $-\infty$. Our results include systematically the noncentered and centered cases of these operators. Here we analyse the Gauss-Weierstrass semigroup on $\mathbb{R}$ as a particular case. We also point out Abelian theorems for other kinds of operators which have been studied in several papers.
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