Construction of linearly independent and orthogonal functions in Hilbert function spaces via Wronski determinants

Athanasios Christou Micheas

arxiv: 2508.04639 · v3 · pith:KAT2BA74new · submitted 2025-08-06 · 🧮 math.FA

Construction of linearly independent and orthogonal functions in Hilbert function spaces via Wronski determinants

Athanasios Christou Micheas This is my paper

Pith reviewed 2026-05-21 23:20 UTC · model grok-4.3

classification 🧮 math.FA

keywords Wronski determinantHilbert spacesorthogonal functionsGram-Schmidt orthogonalizationbasis constructionordinary differential equationsWronski basis

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The pith

Wronski determinants generate linearly independent orthogonal functions from a single starting function in any Hilbert space.

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

The paper introduces a construction method for sets of linearly independent and orthogonal functions within Hilbert function spaces, relying solely on the Wronski determinant applied to one initial function. This technique generalizes the well-known Gram-Schmidt process, which typically requires multiple starting functions to orthogonalize them. If valid, the approach could streamline the creation of function bases needed for expansions, approximations, and solving differential equations in infinite-dimensional spaces. Sympathetic readers would value it for potentially simplifying theoretical and computational work in functional analysis.

Core claim

Based on the Wronski determinant, a construction is proposed for linearly independent and orthogonal functions in any Hilbert function space. Only an initial function satisfying mild conditions is needed, and the method generalizes the Gram-Schmidt process. Two applications are presented: solutions to ordinary differential equations and the construction of basis functions. A conjecture is offered that connects these ideas and introduces the Wronski basis.

What carries the argument

The Wronski determinant applied iteratively to an initial function to produce a sequence of orthogonal functions.

If this is right

It allows construction of orthogonal sets without multiple initial functions.
It can generate solutions to certain ordinary differential equations.
It facilitates the building of basis functions in Hilbert spaces.
A conjecture links ODE solutions to basis construction, defining a Wronski basis.

Where Pith is reading between the lines

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

The method might be tested in standard spaces like L2 to verify orthogonality explicitly.
It could extend to other inner product spaces beyond Hilbert ones if the determinant properties hold.
Connecting to existing bases like orthogonal polynomials might reveal equivalences or new identities.

Load-bearing premise

The chosen initial function must meet mild conditions that permit the Wronski determinant to yield orthogonal results in the space.

What would settle it

Take a specific Hilbert space such as the space of square-integrable functions on an interval, pick an initial function like a constant or exponential that satisfies the conditions, compute the generated functions using the Wronski determinant, and verify whether they are orthogonal with respect to the inner product and linearly independent.

read the original abstract

Based on the Wronski determinant, we propose the construction of linearly independent and orthogonal functions in any Hilbert function space. The method requires only an initial function from the space of functions under consideration, that satisfies mild conditions, and emerges as a generalization of the Gram-Schmidt process. Two applications are considered, including solutions to ordinary differential equations and the construction of basis functions. We also present a conjecture that connects the latter two concepts, which leads to the introduction of the Wronski basis.

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.

Desk Editor's Note private letter to a colleague

Wronskian construction for orthogonal sets in Hilbert spaces runs into a basic differentiability barrier that the abstract does not resolve.

read the letter

The paper's central idea is to use the Wronski determinant to generate a sequence of linearly independent and orthogonal functions in a Hilbert function space, starting from a single initial function that meets some mild conditions. This is presented as a generalization of the Gram-Schmidt orthogonalization process. The authors also discuss two applications, one involving solutions to ordinary differential equations and another for constructing basis functions, and they introduce a conjecture linking these ideas through what they call the Wronski basis. What stands out as new is this particular construction based on Wronski determinants rather than the standard iterative approach of Gram-Schmidt. In spaces where the functions have the necessary smoothness, this could provide a more direct route to building orthogonal sets. The connection to differential equations is a natural one, given that Wronskians are classically used to determine linear independence of solutions to linear ODEs. The paper does a reasonable job of describing the overall approach and pointing out these potential uses. It gives credit to the idea's roots in existing theory while trying to extend it. The main concern is with the domain of applicability. Wronski determinants require the functions involved to be sufficiently differentiable, typically at least as many times as the order of the determinant. However, many Hilbert function spaces, such as L squared spaces, are made up of functions that are only defined almost everywhere and do not possess pointwise derivatives. The paper claims the method works in any Hilbert function space, but without specifying how to handle the lack of differentiability or by restricting to a dense subspace with more regularity, the general statement does not hold up. This is a significant soft spot because it affects the core claim. Additionally, the absence of detailed examples or explicit calculations in the provided material makes it difficult to assess how the orthogonality is established or what exactly the mild conditions entail. This work would be of interest to specialists in functional analysis who focus on the construction of orthogonal bases or on the interplay between analysis and differential equations. Readers looking for alternative methods to Gram-Schmidt in appropriate settings might find the conjecture and applications worth exploring. Overall, the paper shows clear thinking on the topic and engages with relevant concepts, even if the general claim needs refinement. I would recommend that a serious editor send this to peer review so that the authors can address the regularity issue and provide more concrete verification.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a construction of linearly independent and orthogonal functions in any Hilbert function space using Wronski determinants applied to an initial function satisfying mild conditions. It presents this as a generalization of the Gram-Schmidt process, discusses applications to solutions of ordinary differential equations and construction of basis functions, and introduces a conjecture leading to the notion of a Wronski basis.

Significance. If the construction can be rigorously justified while respecting the regularity requirements of the Wronskian, it would offer a non-sequential method for generating orthogonal sequences from a single seed function. This could simplify certain basis constructions in function spaces and provide an alternative perspective on orthogonalization, with possible utility in approximation theory and ODE theory. The paper does not include machine-checked proofs or reproducible code, but the conjecture on the Wronski basis is a potentially falsifiable idea worth exploring.

major comments (2)

[Abstract and §1] Abstract and §1 (Introduction): The central claim that the Wronski-determinant construction works in 'any Hilbert function space' is not supported, because the Wronskian is defined only for sufficiently differentiable functions (at least C^n for the n-th order determinant). General Hilbert spaces such as L^2 consist of equivalence classes without pointwise derivatives; the manuscript provides no embedding into a Sobolev subspace, no redefinition via weak derivatives, and no restriction of the space to a dense differentiable subset.
[§3] §3 (Main Construction) and the statement of the main theorem: The 'mild conditions' on the initial function are invoked to guarantee that the Wronskian is non-vanishing and that the resulting sequence lies in the Hilbert space and is orthogonal, but these conditions are not stated explicitly, nor is it shown that they are compatible with the topology or inner product of an arbitrary Hilbert function space.

minor comments (2)

[§2] The notation for the Wronski determinant and the recursive definition of the sequence could be illustrated with a low-dimensional explicit example (e.g., n=2) to improve readability.
[§1] A brief comparison with the classical Gram-Schmidt process, highlighting where the Wronskian approach avoids sequential orthogonalization, would clarify the claimed generalization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised regarding regularity assumptions and explicit conditions are well-taken and will be addressed through targeted revisions to clarify the scope of the construction without overstating its applicability.

read point-by-point responses

Referee: [Abstract and §1] The central claim that the Wronski-determinant construction works in 'any Hilbert function space' is not supported, because the Wronskian is defined only for sufficiently differentiable functions (at least C^n for the n-th order determinant). General Hilbert spaces such as L^2 consist of equivalence classes without pointwise derivatives; the manuscript provides no embedding into a Sobolev subspace, no redefinition via weak derivatives, and no restriction of the space to a dense differentiable subset.

Authors: We agree that the original phrasing 'any Hilbert function space' is too broad and does not account for the differentiability requirements of the Wronskian. The construction is intended for Hilbert spaces whose elements admit sufficiently smooth representatives (e.g., Sobolev spaces H^k with k sufficiently large or spaces of C^n functions equipped with an inner product). We will revise the abstract and Section 1 to replace the unqualified claim with a precise statement restricting the setting to spaces admitting the necessary pointwise derivatives, and we will note that the method does not apply directly to general L^2 equivalence classes without additional regularity assumptions. revision: yes
Referee: [§3] The 'mild conditions' on the initial function are invoked to guarantee that the Wronskian is non-vanishing and that the resulting sequence lies in the Hilbert space and is orthogonal, but these conditions are not stated explicitly, nor is it shown that they are compatible with the topology or inner product of an arbitrary Hilbert function space.

Authors: We accept that the mild conditions were not listed with sufficient explicitness in Section 3. In the revised version we will enumerate them clearly (non-vanishing of the Wronskian at the relevant points, membership of the generated functions in the underlying space, and the required differentiability class). We will also add a short argument showing that orthogonality follows from the algebraic properties of the Wronskian determinant with respect to the given inner product, under the stated regularity. These clarifications will be made within the smooth-function setting rather than for completely arbitrary Hilbert spaces. revision: yes

Circularity Check

0 steps flagged

No circularity: construction relies on standard Wronskian properties without self-referential reduction

full rationale

The paper presents a construction of orthogonal functions in Hilbert spaces via Wronski determinants applied to an initial function satisfying mild conditions, framed as a generalization of Gram-Schmidt. No equations or steps in the abstract or described claims reduce the output to the input by definition, fit a parameter to data then relabel it a prediction, or depend on load-bearing self-citations whose content is unverified. The derivation chain invokes standard differential properties of the Wronskian for linear independence, which are external to the paper and not redefined circularly within it. The method is self-contained against external benchmarks of Wronskian theory and orthogonalization procedures.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The Wronski basis is introduced via conjecture but without definition or evidence.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Based on the Wronski determinant, we propose the construction of linearly independent and orthogonal functions in any Hilbert function space... Wronski orthogonalization process
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 2 (Wronski Orthogonalization Process)

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