ABSTRACT RELATIVE FOURIER TRANSFORMS OVER CANONICAL HOMOGENEOUS SPACES OF SEMI-DIRECT PRODUCT GROUPS WITH ABELIAN NORMAL FACTOR
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초록
This paper presents a systematic study for theoretical aspects of a unified approach to the abstract relative Fourier transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor. Let $H$ be a locally compact group, $K$ be a locally compact Abelian (LCA) group, and $\theta:H\to {\rm Aut}(K)$ be a continuous homomorphism. Let $G_\theta=H\ltimes_\theta K$ be the semi-direct product of $H$ and $K$ with respect to $\theta$ and $G_\theta/H$ be the canonical homogeneous space (left coset space) of $G_\theta$. We introduce the notions of relative dual homogeneous space and also abstract relative Fourier transform over $G_\theta/H$. Then we study theoretical properties of this approach.
Abstract
This paper presents a systematic study for theoretical aspects of a unified approach to the abstract relative Fourier transforms over canonical homogeneous spaces of semi-direct product groups with Abelian normal factor. Let $H$ be a locally compact group, $K$ be a locally compact Abelian (LCA) group, and $\theta:H\to {\rm Aut}(K)$ be a continuous homomorphism. Let $G_\theta=H\ltimes_\theta K$ be the semi-direct product of $H$ and $K$ with respect to $\theta$ and $G_\theta/H$ be the canonical homogeneous space (left coset space) of $G_\theta$. We introduce the notions of relative dual homogeneous space and also abstract relative Fourier transform over $G_\theta/H$. Then we study theoretical properties of this approach.
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