# The Banach Algebra of Functions With Fourier Transforms in Weighted Amalgam Spaces

## TL;DR

This paper introduces a new Banach algebra of functions with Fourier transforms in weighted amalgam spaces, exploring its algebraic, topological, and functional properties, including modules, embeddings, and multipliers.

## Contribution

It defines a novel function space with Fourier transforms in weighted amalgam spaces and investigates its algebraic and analytical properties, including Banach algebra structure and multipliers.

## Key findings

- The space forms a Banach algebra under convolution.
- The space is translation invariant and a Banach module.
- The paper characterizes multipliers and embedding relations.

## Abstract

In this paper, we define $A_{\vartheta _{1},\vartheta _{2}}^{p,1,q,r}\left(G\right) $ to be space of all functions in $\left( L_{\vartheta_{1}}^{p},\ell ^{1}\right) $ whose Fourier transforms belong to $\left( L_{\vartheta _{2}}^{q},\ell ^{r}\right) .$ Moreover, we consider the basic and advance properties of this space including Banach algebra, translation invariant, Banach module, a generalized type of Segal algebra etc. Also, we study some inclusions, compact embeddings in sense to weights and further discuss multipliers of this space.