# New properties of grand amalgam spaces

## TL;DR

This paper explores new properties of grand amalgam spaces, including an equivalent discrete definition and conditions under which these spaces form an algebra under convolution on locally compact Abelian groups.

## Contribution

It introduces a discrete definition of grand amalgam spaces and characterizes when these spaces are closed under convolution on Abelian groups.

## Key findings

- Established an equivalent discrete definition of grand amalgam spaces.
- Identified necessary and sufficient conditions for these spaces to be convolution algebras.
- Extended the understanding of algebraic structures within grand amalgam spaces.

## Abstract

In $ \left[14\right]$, a new family called grand amalgam space $W( L^{p),\theta},L^{q),\theta })$ of amalgam spaces was defined and investigated properties of these spaces. The present paper is a sequel to my work $[14].$ In this paper, notations are included in Section 1. In Section 2, we introduce another equivalent but discrete definition of grand amalgam space and study properties of these spaces. In Section 3, we determine necessary and sufficient conditions on a locally compact Abelian group $G$ for the grand amalgam spaces $W(L^{p,)\theta},L^ {q,\theta})$ to be an algebra under convolution.