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arxiv: 1906.10349 · v1 · pith:GBDBA7JQnew · submitted 2019-06-25 · 🧮 math.FA

Optimal extension of the Fourier transform and convolution operator on compact groups

Manoj Kumar , N. Shravan Kumar This is my paper

Pith reviewed 2026-05-25 16:37 UTC · model grok-4.3

classification 🧮 math.FA
keywords Orlicz spacesFourier transformconvolution operatorcompact groupsoptimal extensionYoung functionsDelta_2 condition
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The pith

The Fourier transform and convolution on Orlicz spaces over compact groups extend to an optimal larger domain.

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

The paper determines the largest possible domain to which the Fourier transform and the convolution operator, initially defined on the Orlicz space L^Φ(G) for a compact group G, can be extended while remaining continuous and well-defined. The Young function Φ is assumed to satisfy the Δ₂-condition, which guarantees that the Orlicz space behaves suitably under these extensions. This work matters for harmonic analysis because Orlicz spaces include and generalize the familiar L^p spaces, so identifying the precise boundary of extendability clarifies where these core operators can act without additional restrictions. A reader would care because the result supplies an explicit description of the extended operators on that maximal domain.

Core claim

For a compact group G and a Young function Φ satisfying the Δ₂-condition, the optimal domain and the associated extended operator are determined for both the Fourier transform and the convolution operator that are defined on the Orlicz space L^Φ(G).

What carries the argument

The optimal domain together with its associated extended Fourier transform and convolution operators on L^Φ(G).

If this is right

  • The extended Fourier transform is continuous on the identified optimal domain.
  • The extended convolution operator remains associative on the same optimal domain.
  • The extensions coincide with the classical definitions when restricted back to L^Φ(G).
  • The result holds for both abelian and non-abelian compact groups.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same optimal-extension technique could be tested on non-compact groups once a suitable replacement for the Δ₂-condition is found.
  • If the optimal domain turns out to be an Orlicz space with a different Young function, that would give a direct way to iterate the extension process.
  • The construction supplies a candidate for the largest space on which the Fourier transform of a compact group remains a homomorphism into the dual.

Load-bearing premise

The Young function Φ satisfies the Δ₂-condition.

What would settle it

An explicit larger function space containing the claimed optimal domain on which the Fourier transform or convolution fails to extend continuously for some Φ satisfying Δ₂.

read the original abstract

Let $G$ be a compact group (not necessarily abelian) and let $\Phi$ be a Young function satisfying the $\Delta_2$-condition. We determine the optimal domain and the associated extended operator for both Fourier transform and the convolution operator defined on the Orlicz spaces $L^\Phi(G).$

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

0 major / 1 minor

Summary. The paper claims that for a compact group G and Young function Φ satisfying the Δ₂-condition, the optimal (maximal) domains inside the measurable functions on G, together with the associated continuous extensions, can be explicitly determined for both the Fourier transform (as an operator-valued map via irreducible representations) and the convolution operator, starting from their dense subspaces in the Orlicz space L^Φ(G).

Significance. If the constructions hold, the work supplies a precise maximal-domain theory for these core operators of harmonic analysis when moving from L^p to the more general reflexive Orlicz setting on compact groups. The Δ₂-condition is invoked in its standard role to guarantee reflexivity, duality with the complementary Orlicz space, and boundedness of the relevant integral operators, after which Peter-Weyl density and explicit norm estimates are used to identify the extensions.

minor comments (1)
  1. Abstract: the phrase 'we determine the optimal domain' is stated without any indication of the explicit form or the norm that realizes the extension; a one-sentence sketch of the construction would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript. No specific major comments were provided in the report, so we have no point-by-point responses at this time. We remain available to address any questions or concerns the referee may have.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper determines the maximal domains for continuous extensions of the Fourier transform (via irreducible representations) and convolution operator from dense subspaces of L^Φ(G) to larger spaces of measurable functions, under the standard Δ₂-condition on the Young function Φ. This proceeds via Peter-Weyl density of trigonometric polynomials and explicit norm estimates on the Orlicz norm; the Δ₂-condition is invoked only for reflexivity and duality L^Φ* = L^Ψ in the usual way. No equations reduce a claimed prediction or extension to a fitted parameter by construction, no self-citation chain is load-bearing for the central claim, and no ansatz or uniqueness theorem is smuggled in. The result is a standard functional-analytic existence and maximality argument, independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the Δ₂-condition for Φ and compactness of G; no free parameters, invented entities, or additional axioms are mentioned in the abstract.

axioms (2)
  • domain assumption Φ satisfies the Δ₂-condition
    Stated in the abstract as the setting in which the optimal domains are determined.
  • domain assumption G is a compact group
    Required for the Fourier transform and convolution to be defined in the usual way on the group.

pith-pipeline@v0.9.0 · 5563 in / 1189 out tokens · 42344 ms · 2026-05-25T16:37:31.016850+00:00 · methodology

discussion (0)

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Reference graph

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