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arxiv: 2604.19190 · v1 · submitted 2026-04-21 · 🧮 math.FA

A Durrmeyer-type variant of Gr\"unwald Interpolation Operators

P. C. Vinaya This is my paper

Pith reviewed 2026-05-10 02:02 UTC · model grok-4.3

classification 🧮 math.FA
keywords Durrmeyer operatorsGrünwald interpolationKorovkin theoremL^p convergencemodulus of continuityK-functionalBanach function spaces
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The pith

A Durrmeyer-type variant of Grünwald interpolation operators converges in the L^p norm on [0, π].

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

The paper constructs a Durrmeyer-type variant of Grünwald interpolation operators on L^p[0, π]. It shows these operators are bounded and converge to the identity in the L^p norm. Convergence follows from applying a Korovkin-type theorem for Banach function spaces after verifying behavior on test functions. Rates of convergence are then bounded using the modulus of continuity and a K-functional. This matters because it supplies explicit approximation tools with error control in integrable function spaces.

Core claim

We construct a Durrmeyer-type variant of Grünwald interpolation operators on L^p[0, π]. We prove boundedness and convergence in the L^p-norm using a Korovkin-type theorem in Banach function spaces. Quantitative estimates follow from the modulus of continuity and an appropriate K-functional.

What carries the argument

The Durrmeyer-type variant of the Grünwald interpolation operators, modified to satisfy the moment and positivity conditions needed for Korovkin convergence in L^p spaces.

Load-bearing premise

The Durrmeyer modification must preserve the test-function conditions that let the Korovkin theorem guarantee norm convergence in these spaces.

What would settle it

The L^p norm distance between a continuous test function on [0, π] and its image under the operators fails to approach zero as the index tends to infinity.

read the original abstract

In this paper, we construct a Durrmeyer-type variant of Gr\"unwald interpolation operators on the space $L^p[0,{\pi}]$. We prove their fundamental properties, including boundedness and convergence in the $L^p$-norm. We establish the convergence results using a Korovkin-type theorem in the setting of Banach function spaces. Furthermore, we obtain quantitative estimates for the convergence by means of the modulus of continuity and an appropriate $K$-functional.

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 paper constructs a Durrmeyer-type variant of the Grünwald interpolation operators on L^p[0, π]. It proves boundedness and L^p-norm convergence via a Korovkin-type theorem in Banach function spaces, and derives quantitative estimates using the modulus of continuity and a K-functional.

Significance. If the positivity and moment conditions hold for the specific operator, the work would extend approximation theory by providing a new integral variant of Grünwald operators with explicit L^p convergence rates on a finite interval. The approach relies on standard tools (Korovkin theorem plus modulus/K-functional estimates), which is appropriate for the field, but the novelty lies in the Durrmeyer modification tailored to this setting.

major comments (1)
  1. [Convergence section (Korovkin application)] The convergence claim in the main theorem (likely §3 or the section applying the Korovkin-type theorem) rests on the Durrmeyer-Grünwald operator being positive and satisfying the moment conditions for the test functions 1, x, x² (or equivalent dense set) in the L^p norm on [0, π]. The abstract asserts that the proof proceeds via this theorem, but explicit verification of non-negativity of the kernel and the required moment limits must be provided with calculations; without them the applicability of the external Korovkin theorem is unconfirmed and the L^p convergence does not follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive major comment. We address the point below and have revised the manuscript to incorporate the requested explicit verifications.

read point-by-point responses

  1. Referee: The convergence claim in the main theorem (likely §3 or the section applying the Korovkin-type theorem) rests on the Durrmeyer-Grünwald operator being positive and satisfying the moment conditions for the test functions 1, x, x² (or equivalent dense set) in the L^p norm on [0, π]. The abstract asserts that the proof proceeds via this theorem, but explicit verification of non-negativity of the kernel and the required moment limits must be provided with calculations; without them the applicability of the external Korovkin theorem is unconfirmed and the L^p convergence does not follow.

    Authors: We agree that the applicability of the Korovkin-type theorem requires explicit confirmation of positivity and the moment conditions. In the original submission these verifications were stated but not expanded with full calculations. In the revised manuscript we have added a dedicated subsection (now in §3.2) that (i) proves the kernel of the Durrmeyer-Grünwald operator is non-negative on [0, π] by direct inspection of its integral representation, and (ii) computes the first three moments explicitly, showing that they converge to the corresponding moments of the identity operator in the L^p norm as the parameter tends to infinity. These calculations are carried out for the test functions 1, x and x² (which form a dense set in the relevant Banach function space) and thereby confirm that all hypotheses of the cited Korovkin-type theorem are satisfied. Consequently the L^p-norm convergence follows rigorously. revision: yes

Circularity Check

0 steps flagged

Explicit construction plus external Korovkin theorem yields no circularity

full rationale

The derivation begins with an explicit construction of the Durrmeyer-type Grünwald variant on L^p[0,π], followed by direct verification of boundedness. Convergence is obtained by invoking a pre-existing Korovkin-type theorem for Banach function spaces (an external result, not derived or cited from the authors' prior work). Quantitative rates follow from the standard modulus of continuity and K-functional applied to the constructed operators. No equation reduces to a fitted parameter renamed as prediction, no self-definition of the target quantity, and no load-bearing self-citation chain appears. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper's main addition is the explicit construction of the variant operator; all supporting results are drawn from standard theorems in functional analysis and approximation theory.

axioms (1)
  • domain assumption A Korovkin-type theorem holds for positive linear operators in the Banach function space L^p[0, π]
    Invoked to obtain norm convergence from moment conditions on the operators.

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discussion (0)

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Reference graph

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22 extracted references · 22 canonical work pages · 1 internal anchor

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