  Let $G$ be a non-discrete LCA group with the dual group $\Gamma$. We define
generalized group algebra, ${\mathcal L}^1(G)$, and generalized measure
algebra, ${\mathcal M}(G),$ on $G$ as generalizations of the group algebra
$L^1(G)$ and the measure algebra $M(G)$, respectively. Generalized Fourier
transforms of elements of ${\mathcal L}^1(G)$ and generalized Fourier-Stieltjes
transforms of elements of ${\mathcal M}(G)$ are also defined as generalizations
of the Fourier transforms and the Fourier-Stieltjes transforms, respectively.
The image ${\mathcal A}(\Gamma)$ of ${\mathcal L}^1(G)$ by the generalized
Fourier transform becomes a function algebra on $\Gamma$ with norm inherited
from ${\mathcal L}^1(G)$ through this transform. It is shown that ${\mathcal
A}(\Gamma)$ is a natural Banach function algebra on\, $\Gamma$\,which is BSE
and BED. It turns out that ${\mathcal L}^1(G)$ contains all Rajchman measures.
Segal algebras in ${\mathcal L}^1(G)$ are defined and investigated. It is shown
that there exists the smallest isometrically character invariant Segal algebra
in ${\mathcal L}^1(G)$, which (eventually) coincides with the smallest
isometrically character invariant Segal algebra in $L^1(G)$, the Feichtinger
algebra of $G$. A notion of locally bounded elements of ${\mathcal M}(G)$ is
introduced and investigated. It is shown that for each locally bounded element
$\mu$ of ${\mathcal M}(G)$ there corresponds a unique Radon measure $\iota \mu$
on $G$ which characterizes $\mu$. We investigate the multiplier algebra
$\mathbb{M}({\mathcal L}^1(G))$ of ${\mathcal L}^1(G)$, and obtain a result
that there is a natural continuous isomorphism from $\mathbb{M} ({\mathcal
L}^1(G))$ into $A(G)^*$, the algebra of pseudomeasures on $G$. When $G$ is
compact, this map becomes surjective and isometric.
