Homogeneous Banach spaces as Banach convolution modules over $M(G)$

Abstract: We develop an elementary approach to convolution and Fourier analysis on a locally compact Abelian group (G), based on bounded measures and bounded uniform partitions of unity. In earlier work, the author introduced convolution and the Fourier--Stieltjes transform on the Banach space (M(G)) of bounded measures, viewed as linear functionals, in a direct Euclidean setting. The present paper constructs arbitrarily fine bounded uniform partitions of unity on general locally compact Abelian groups. The construction is designed to avoid structure theory and does not presuppose Haar measure or Lebesgue integration. It is then used to establish a natural convolution-module structure of (M(G)) on a broad class of homogeneous Banach spaces on (G). This class includes (L^p(G)), for (1\leq p<\infty), the Fourier--Stieltjes algebra, and, in particular, Segal algebras. After introducing Haar measure, we identify (L^1(G)) with the closure in (M(G)) of the measure-embedded space (C_c(G)). We prove that the homogeneous Banach spaces under consideration are essential (L^1(G))-modules. Consequently, standard approximate identities act in the expected manner and converge strongly to the identity operator. The method follows the spirit of Hans Reiter: it avoids the customary reliance on LCA-group structure theory and on vector-valued integration arguments based on duality. It is intended as a foundation for a subsequent elementary treatment of the extended Fourier transform in the Banach Gelfand triple generated by a Segal algebra.

• The paper constructs arbitrarily fine bounded uniform partitions of unity on every locally compact group using a translation-invariant covering capacity, without relying on Haar measure or structure theory.
• The paper extends every strongly continuous isometric representation of a locally compact group to a uniquely determined, associative and bounded convolution action of the measure algebra M(G).
• The paper proves that every homogeneous Banach space is an essential L¹(G)-module, so bounded approximate identities in L¹(G) converge strongly to the identity on each element of the space.

Motivation and program

This paper by H. G. Feichtinger continues a long-term project of rebuilding harmonic analysis on locally compact Abelian (LCA) groups from elementary, operator-theoretic ingredients, deliberately avoiding both the structure theory of LCA groups and the machinery of vector-valued integration via duality. The starting point is the author's earlier observation that the measure algebra $M(G)$ — defined as the space of bounded linear functionals on $C_0(G)$ — can be endowed with convolution and a Fourier–Stieltjes transform in an entirely elementary way, using the isometric identification between $M(G)$ and bounded linear operators on $C_0(G)$ that commute with translations (translation-invariant linear systems, TILS). The present contribution supplies two technical keystones for that program: an explicit construction of arbitrary fine bounded uniform partitions of unity (BUPUs) on any locally compact group without invoking Haar measure or structure theory, and the proof that every homogeneous Banach space (HBS) on $G$ is a Banach convolution module over $M(G)$, and indeed an essential module over $L^1(G)$. The declared ultimate goal — explicitly deferred to subsequent papers — is an elementary construction of the Fourier transform within the Banach Gelfand triple $(S_0(G), L^2(G), S_0'(G))$ built around the Segal algebra $S_0(G)$.

Bounded uniform partitions of unity

The paper first compares several variants of uniform partitions of unity. The concept of choice is the BUPU introduced by Feichtinger in 1983: a family $\Psi = (\psi_i)_{i \in I}$ of continuous functions with $C_0(G)$0 for a neighborhood $C_0(G)$1 of the identity, satisfying the bounded overlap property (BOP) with overlap bound $C_0(G)$2, and summing identically to one. A simplified non-negative version suffices here; a more general version, bounded in a pointwise Banach algebra $C_0(G)$3-type algebra, is needed for Wiener amalgam spaces. The paper contrasts these with the UPUs of Leptin and Müller [lemu91], where all constituents are translates of a single function $C_0(G)$4. Leptin–Müller prove existence of UPUs on any locally compact group, but their argument relies heavily on structure theory, and it remains open whether arbitrary fine UPUs exist in that sense. The BUPU concept is more flexible, and fineness is exactly what the later module results require.

The historical remarks connect BUPUs to quasi-interpolation operators, Wiener amalgam norms, coorbit theory with Gröchenig, and iterative reconstruction of band-limited functions from irregular samples — contexts where BUPUs centered at prescribed $C_0(G)$5-dense families are essential.

Arbitrary fine BUPUs via a capacity

The first main result constructs arbitrary fine non-negative BUPUs on any locally compact group using only topology. For a relatively compact reference set $C_0(G)$6, a "capacity" is defined on compact sets by

$C_0(G)$7

i.e., the minimal number of translates of $C_0(G)$8 covering $C_0(G)$9. This is a coarse substitute for Haar measure: it satisfies $M(G)$0, translation invariance, subadditivity, and additivity on $M(G)$1-separated families. The author notes the analogy with the classical construction of the Haar measure (Weil) and leaves to the reader the verification that $M(G)$2 can be read as an infimum of norms of functions dominating $M(G)$3.

The construction itself follows the standard pavement pattern: choose symmetric neighborhoods with $M(G)$4, $M(G)$5; take a maximal $M(G)$6-separated family $M(G)$7 so that $M(G)$8 covers $M(G)$9; pick $C_0(G)$0 equal to 1 on $C_0(G)$1 with support in $C_0(G)$2; then $C_0(G)$3 satisfies the sharp pointwise bound

$C_0(G)$4

and normalizing gives $C_0(G)$5, a partition of unity of size $C_0(G)$6 with uniformly separated centers. The finite-overlap control comes precisely from the additivity property of the capacity applied to the $C_0(G)$7-separated family $C_0(G)$8. The significance is foundational: the integrated group action can be established before any measure theory is available, since neither Lebesgue integration nor the Haar measure enters the argument.

Integrated representations of $C_0(G)$9

The second main result operates at the level of abstract homogeneous Banach spaces: a Banach space $G$0 carrying a strongly continuous, isometric representation $G$1. Discrete measures act naturally via absolutely convergent sums, with the norm estimate $G$2, and associativity holds on discrete measures because it holds on Dirac measures. For general $G$3, the action is defined as the limit of Riemann-type sums $G$4 along joint refinements of BUPUs; completeness of $G$5 guarantees convergence, and the key refinement estimate bounds $G$6 by $G$7, which tends to zero by strong continuity. Associativity for general measures then follows by density of discrete measures in the $G$8-sense.

Two remarks qualify the scope. First, the result extends verbatim to projective representations (e.g., the Schrödinger representation relevant to time-frequency analysis), where multiplicativity holds only up to a unimodular constant. Second, if the representation is not isometric but weighted-bounded, one must pass to Beurling algebras; this extension, relevant to translation-modulation invariant Banach spaces (TMIBs), is deferred to future work. The paper also records a cautionary counterexample due to H. C. Wang: a "semi-homogeneous" Banach space lacking strong continuity admits no such extension, confirming that strong continuity is not dispensable. It further observes that Shapiro made associativity an extra axiom in his treatment, apparently unable to derive it directly — here it is obtained as a theorem.

The Wiener algebra and homogeneous Banach spaces without Haar measure

Before invoking the Haar measure, the paper characterizes the Wiener algebra $G$9 on any LCA group via any BUPU: it is a Banach ideal in $M(G)$0 under pointwise multiplication, independent of the chosen BUPU up to equivalent norms, admits an atomic characterization via absolutely convergent sums of translates supported in a fixed compact neighborhood, and has an isometrically translation-invariant inf-norm with continuous translations. Its dual $M(G)$1 is the space of translation-bounded measures, characterized by a uniform estimate $M(G)$2 on atoms supported in translates of $M(G)$3, and admitting an atomic $M(G)$4-convergent representation $M(G)$5 with bounded $M(G)$6. These descriptions require no Haar integral.

This prepares a Haar-free definition of a homogeneous Banach space on an LCA group: continuous embedding into $M(G)$7, isometric and strongly continuous translations. The paper shows (via the Cohen–Hewitt factorization theorem and Wiener amalgam convolution relations) that every such HBS embeds continuously into $M(G)$8 once the usual Lebesgue setting is available, reconciling the abstract definition with Katznelson's classical one.

Essential $M(G)$9-module structure

With the Haar measure now invoked, $L^1(G)$0 is realized as the closure of $L^1(G)$1 inside $L^1(G)$2 under the isometric embedding $L^1(G)$3. The paper establishes that $L^1(G)$4 is a closed ideal in $L^1(G)$5 and $L^1(G)$6-dense in it, the latter via explicit approximation of Dirac measures by normalized bump functions. Any Segal algebra (a densely and continuously embedded HBS in $L^1(G)$7) is a Banach ideal in $L^1(G)$8.

The third main result strengthens the module picture considerably: for a bounded and tight net $L^1(G)$9 in $(S_0(G), L^2(G), S_0'(G))$0 converging $(S_0(G), L^2(G), S_0'(G))$1 to $(S_0(G), L^2(G), S_0'(G))$2, the corresponding operators converge strongly,

$(S_0(G), L^2(G), S_0'(G))$3

The proof combines three estimates of size $(S_0(G), L^2(G), S_0'(G))$4: discretization error controlled by BUPU fineness, tightness reducing to jointly compactly supported measures, and $(S_0(G), L^2(G), S_0'(G))$5-convergence on the finitely many remaining atoms. An immediate consequence is that bounded approximate units (Dirac sequences) in $(S_0(G), L^2(G), S_0'(G))$6 act correctly:

$(S_0(G), L^2(G), S_0'(G))$7

so every HBS is an essential Banach module over $(S_0(G), L^2(G), S_0'(G))$8. A companion corollary verifies the full associativity law $(S_0(G), L^2(G), S_0'(G))$9 via the iteration principle for convergent nets. The final structural theorem states that the restriction of the $S_0(G)$0-action to $S_0(G)$1 yields the essential module structure, and conversely that $S_0(G)$2-continuity on bounded tight families forces the $S_0(G)$3-action to be uniquely determined by the integrated group action. The author argues that obtaining the integrated representation of $S_0(G)$4 by restriction from $S_0(G)$5 is technically simpler than extending the $S_0(G)$6-action by vague limits — reversing the customary order of exposition.

Technical apparatus: Cauchy nets

A short self-contained section records that Banach completeness is equivalent to convergence of Cauchy nets, not merely sequences. The proof extracts a rapidly convergent sequence from the net via a telescoping absolutely convergent series; the author flags the delicate point that the resulting sequence need not be a subnet in the strict topological sense, though subnet terminology is not required. This matters because the Riemann-type sums defining $S_0(G)$7 form nets indexed by BUPU refinements, not sequences.

Limitations and open questions

The paper is candid about its boundaries. The existence of arbitrary fine UPUs in the stricter Leptin–Müller sense (single-generator partitions) remains unresolved for general locally compact groups. The weighted, non-isometric case requires Beurling algebras and is only sketched. The projective-representation extension and the TMIB applications are announced for subsequent work. Most significantly, the stated goal — an elementary construction of the Fourier transform on the triple $S_0(G)$8 for general LCA groups — is explicitly described as "still far," and the present results provide only the module-theoretic foundation. Whether the capacity-based methods can replace Haar-measure arguments throughout the Segal algebra theory is left implicit rather than proven.

Conclusion

The paper delivers three concrete results: a structure-theory-free construction of arbitrary fine BUPUs on any locally compact group via a combinatorial capacity; the extension of any strongly continuous isometric group representation to a uniquely determined, associative Banach module action of $S_0(G)$9; and the identification of homogeneous Banach spaces as essential $\Psi = (\psi_i)_{i \in I}$0-modules whose approximate units converge strongly to the identity. Together they justify treating the measure algebra, rather than $\Psi = (\psi_i)_{i \in I}$1, as the primary object through which convolution acts on function spaces, and they lay the groundwork for the author's planned Haar-free development of the Fourier transform in the $\Psi = (\psi_i)_{i \in I}$2-based Banach Gelfand triple.