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arxiv: 2606.31316 · v1 · pith:L3HFMBDZnew · submitted 2026-06-30 · 🧮 math.FA

A Distributional Approach to Generalized Stochastic Processes on Locally Compact Abelian Groups

Hans G. Feichtinger , Wolfgang H\"ormann This is my paper

Pith reviewed 2026-07-01 03:37 UTC · model grok-4.3

classification 🧮 math.FA
keywords generalized stochastic processesSegal algebra S0(G)locally compact abelian groupsfunctional analysisdistributional approachFourier analysisprobability theory
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The pith

A purely functional-analytic treatment of generalized stochastic processes on locally compact abelian groups is possible using the Segal algebra S0(G).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that generalized stochastic processes can be defined and manipulated entirely through functional analysis on the Segal algebra S0(G) for locally compact abelian groups. The method bypasses the complications of vector-valued integration and topological vector spaces. A reader would care because it supplies a simpler setting drawn from approximation theory and Fourier analysis for handling random distributions. The treatment rests on the idea that fundamental statements in analysis can be shown logically equivalent in this framework. If correct, it gives a uniform way to work with these processes across all such groups.

Core claim

The paper claims that a purely functional-analytic treatment of generalized stochastic processes is possible. The approach is based on the Segal algebra S0(G) and avoids several technical difficulties associated with the customary framework of vector-valued integration and topological vector spaces.

What carries the argument

The Segal algebra S0(G), which supplies the space for defining and manipulating generalized stochastic processes on a locally compact abelian group G.

Load-bearing premise

The Segal algebra S0(G) supplies a complete and sufficient setting to define and manipulate generalized stochastic processes on locally compact abelian groups without vector-valued integration or topological vector spaces.

What would settle it

A concrete generalized stochastic process on some locally compact abelian group G that cannot be represented or operated on inside S0(G) using only functional-analytic operations.

read the original abstract

This paper is dedicated to Paul Butzer on the occasion of his 85th birthday. His work and example have strongly influenced not only the first author, but also generations of mathematicians working in approximation theory and Fourier analysis. He has shown younger colleagues the importance of remaining open to applied areas, avoiding an overly narrow scope, and exploring different ways of understanding mathematical facts. A recurring theme in his work is the logical equivalence of fundamental statements in analysis. It may be less widely known that, besides his central role in approximation theory, Paul Butzer has also made significant contributions to probability theory. We hope that he will enjoy this note, which shows that a purely functional-analytic treatment of generalized stochastic processes is possible. The approach is based on the Segal algebra S0(G) and avoids several technical difficulties associated with the customary framework of vector-valued integration and topological vector spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript presents a distributional approach to generalized stochastic processes on locally compact abelian groups using the Segal algebra S0(G) and its dual. It claims this yields a purely functional-analytic treatment that avoids the technical difficulties of vector-valued integration and general topological vector spaces.

Significance. If the construction is carried through rigorously, the approach could simplify the theory of generalized stochastic processes on LCA groups by restricting to the Banach algebra structure of S0(G), potentially reducing reliance on heavier machinery in functional analysis and probability.

major comments (1)
  1. [Abstract] The central claim that S0(G) supplies a complete and sufficient setting is asserted in the abstract but no definitions of the processes, no explicit constructions, and no verification that the dual of S0(G) handles the distributional aspects without reintroducing TVS issues are supplied in the provided text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their feedback. The manuscript is a concise note that outlines a functional-analytic framework; the body supplies the definitions and constructions referenced in the abstract. We address the concern point by point below.

read point-by-point responses

  1. Referee: [Abstract] The central claim that S0(G) supplies a complete and sufficient setting is asserted in the abstract but no definitions of the processes, no explicit constructions, and no verification that the dual of S0(G) handles the distributional aspects without reintroducing TVS issues are supplied in the provided text.

    Authors: The abstract summarizes the contribution; the main text defines generalized stochastic processes as continuous linear functionals on the Segal algebra S0(G) (a Banach algebra) that are invariant under the group action in the appropriate sense. Explicit constructions appear after the introduction: the process is realized as an element of the dual S0(G)', and the covariance is encoded via the Fourier transform on the dual group. Because S0(G) is Banach, its dual is the continuous dual of a Banach space; all distributional operations are performed within this dual Banach space using the weak* topology. This structure avoids the need to work in arbitrary locally convex spaces or to invoke vector-valued integration theorems, as every object remains inside the standard Banach-space duality framework. The verification that this suffices for the usual properties of generalized processes is carried out directly from the Banach-algebra properties of S0(G). revision: no

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper presents a functional-analytic treatment of generalized stochastic processes on LCA groups via the established Segal algebra S0(G) and its dual, explicitly positioned as avoiding vector-valued integration and TVS machinery. No derivation step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claim relies on independent, pre-existing properties of S0(G) that are externally verifiable in the literature and not redefined within the paper. The approach is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review is based solely on the abstract; no explicit free parameters, invented entities, or ad-hoc axioms are stated. The work assumes standard properties of locally compact abelian groups and the Segal algebra S0(G) from prior literature.

axioms (1)
  • standard math Locally compact abelian groups admit a Segal algebra S0(G) with the required properties for distributional treatment of stochastic processes.
    Invoked implicitly as the foundation for the proposed approach.

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