Homogeneous Banach Spaces as Banach Convolution Modules over M(G)

Hans Georg Feichtinger

doi:10.3390/math10030364

Abstract

This paper is supposed to form a keystone towards a new and alternative approach to Fourier analysis over LCA (locally compact Abelian) groups G. In an earlier paper the author has already shown that one can introduce convolution and the Fourier–Stieltjes transform on (M(G),∥·∥M), the space of bounded measures (viewed as a space of linear functionals) in an elementary fashion over Rd. Bounded uniform partitions of unity (BUPUs) are easily constructed in the Euclidean setting (by dilation). Moving on to general LCA groups, it becomes an interesting challenge to find ways to construct arbitrary fine BUPUs, ideally without the use of structure theory, the existence of a Haar measure and even Lebesgue integration. This article provides such a construction and demonstrates how it can be used in order to show that any so-called homogeneous Banach space(B,∥·∥B) on G, such as (Lp(G),∥·∥p), for 1≤p<∞, or the Fourier–Stieltjes algebra FM(G), and in particular any Segal algebra is a Banach convolution module over (M(G),∥·∥M) in a natural way. Via the Haar measure we can then identify L1(G),∥·∥1 with the closure (of the embedded version) of Cc(G), the space of continuous functions with compact support, in (M(G),∥·∥M), and show that these homogeneous Banach spaces are essentialL1(G)-modules. Thus, in particular, the approximate units act properly as one might expect and converge strongly to the identity operator. The approach is in the spirit of Hans Reiter, avoiding the use of structure theory for LCA groups and the usual techniques of vector-valued integration via duality. The ultimate (still distant) goal of this approach is to provide a new and elementary approach towards the (extended) Fourier transform in the setting of the so-called Banach–Gelfand triple(S0,L2,S0′)(G), based on the Segal algebra S0(G). This direction will be pursued in subsequent papers.

Highlights

This paper is supposed to form a keystone towards a new and alternative approach to Fourier analysis over locally compact Abelian (LCA) groups G
Let us begin with the observation that the usual approach to harmonic analysis over locally compact Abelian (LCA) groups
Different Types of Uniform Partitions. It is the purpose of this section to compare various notions of uniform partitions of unity in the context of harmonic analysis over LCA groups

Summary

Introduction with regard to jurisdictional claims in

Let us begin with the observation that the usual approach to harmonic analysis over locally compact Abelian (LCA) groups. They allow one to decompose every μ ∈ M ( G ) into an absolutely convergent sum of well-localized measures, which, among other approaches, allows the extension of the action of μ ∈ C00 ( G ) to all of Cb ( G ), the continuous, bounded functions on G ( endowed with the sup-norm) In this way it is possible to define the Fourier–Stieltjes transform of bounded measures and derive the convolution theorem before even discussing the existence of a Haar measure or the necessary Lebesgue integration theory required in order to study everything in the L1 -context. We formulate an even more general abstract approach based on isometric, strongly continuous representations of the group G on an arbitrary Banach space ( B, k · k B ) This approach is based on the methods developed in [5] and makes use of a constructive way of approximating bounded measures by discrete measures in the w∗− sense. In particular it is shown that any homogeneous Banach space is an essential Banach module over L1 ( G ), k · k1

Different Types of Uniform Partitions

Arbitrary Fine BUPUs over LC Groups

Towards Integrated Group Representations

Homogeneous Banach Spaces as Essential L1 -Modules

Findings

Some Basic Functional Analysis