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Legendre compressions and an integrality conjecture for the Hörmander--Bernhardsson extremal function

This paper proves an integrality conjecture regarding the recurrence coefficients of the Hörmander--Bernhardsson extremal function by demonstrating that they are polynomials with integer coefficients derived from the determinants of tridiagonal Legendre compressions, thereby establishing that specific constants related to the function's zeros and the sharp point-evaluation constant for $PW^1$ cannot be simultaneously rational.

Original authors: Khai-Hoan Nguyen-Dang

Published 2026-03-26

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CC BY 4.0

Original authors: Khai-Hoan Nguyen-Dang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery involving a very special, invisible shape called the Hörmander–Bernhardsson extremal function. Let's call this shape $\phi$ (Phi).

This shape is unique. It's the "most efficient" shape possible in a specific mathematical world (called Paley-Wiener space) that satisfies two rules:

It has a specific height at the center (it equals 1 at zero).
It is as "small" as possible overall (it has the minimum total area under its curve).

Mathematicians have been studying this shape for years. They know it has a secret code hidden inside it, written in a language of numbers and variables. The mystery of this paper is about Conjecture 2, a guess made by a team of four mathematicians (Bondarenko, Ortega-Cerdà, Radchenko, and Seip).

The Mystery: The "Magic" Recurrence

The secret code of $\phi$ is generated by a machine that builds numbers one by one. This machine follows a strict recipe called a recurrence relation.

Think of this machine like a baker making a new batch of cookies every day.

Day 0: The baker starts with 1 cookie.
Day 1: To make the next batch, the baker takes the previous batch, mixes in some "flour" (a variable called $\lambda$ ), and adds a pinch of "sugar" (a variable called $x$ ).
Day 2 and beyond: The recipe gets more complex. The baker takes the previous two batches, mixes them with specific amounts of flour and sugar, and adds a "magic multiplier" (a fraction like $\frac{4n+2}{n+1}$ ).

The Problem:
In the recipe, the baker uses fractions. Usually, when you mix fractions, you end up with messy, non-whole numbers (like 3.5 or 7/12).
However, the Conjecture claimed something magical: No matter how many days you bake, the final result is always a whole number (an integer). The fractions always cancel out perfectly, leaving behind clean, whole numbers.

The author of this paper, Khai-Hoan Nguyen-Dang, set out to prove this magic is real.

The Solution: The "Legendre Compression"

How do you prove that fractions always cancel out? You usually try to do the math, but it gets messy. Instead, the author used a clever trick involving Legends.

In math, there are special shapes called Legendre Polynomials. Think of them as a set of perfectly balanced, pre-made building blocks.
The author realized that the "baking machine" (the recurrence relation) wasn't just mixing random ingredients. It was actually performing a compression of these special Legendre blocks.

Here is the analogy:
Imagine you have a giant, messy pile of Lego bricks (the fractions and variables).

The Old Way: Trying to sort the bricks by hand to see if they form a perfect whole number.
The Author's Way: The author realized that if you look at the pile through a special Legendre Lens, the messy pile suddenly snaps together into a neat, solid, rectangular block.

This "block" is a determinant (a specific calculation from a grid of numbers). The author proved that:

The messy recipe the bakers were following is exactly the same as calculating the size of this neat block.
Because the block is built from whole-number ingredients (integers), the result must be a whole number.

So, the "magic" of the fractions canceling out isn't luck; it's because the whole process is just a different way of counting whole Lego blocks.

The Big Discoveries (The "So What?")

Once the author proved that the numbers are always whole, two huge consequences fell out of the bag:

1. The "Irrational" Truth
The paper proves that two very important numbers related to this shape $\phi$ cannot both be "nice" rational numbers (like 1/2 or 3.4).

One number is related to the sharpness of the shape's peak ( $C$ ).
The other is related to the sum of its wiggles ( $L_\tau(1)$ ).

The proof shows that if you try to write both of these as simple fractions, the math breaks. At least one of them must be an irrational number (like $\pi$ or $\sqrt{2}$ ), which goes on forever without repeating. It's like proving you can't build a perfect square out of two specific types of bricks; one of the bricks must be a weird, infinite shape.

2. The "Even Power" Rule
The paper also looked at the ingredients of the shape $\phi$ itself. It turns out that if you write $\phi$ as a sum of powers (like $z^2, z^4, z^6...$ ), the coefficients (the numbers in front) have a very strict rule:

They can contain $\pi$ (pi), but only as $\pi^2, \pi^4$ , etc.
They can never contain just $\pi$ or $\pi^3$ .

It's as if the shape $\phi$ has a "No Odd Pi" policy. The author proved this by showing that the "weight" of the ingredients in the recipe forces the odd powers to disappear.

Summary

In simple terms:

The Puzzle: A complex math recipe seemed to produce whole numbers despite using fractions.
The Trick: The author showed this recipe is secretly just a way of counting whole building blocks (Legendre polynomials).
The Result: The recipe always produces whole numbers.
The Bonus: This proof revealed that two famous mathematical constants related to this shape are "irrational" partners, and that the shape's formula only allows for even powers of Pi.

The author didn't just solve a math problem; they found a hidden "Lego structure" underneath a messy pile of fractions, proving that order exists where chaos seemed to be.

1. Problem Statement and Context

The paper addresses Conjecture 2 proposed by Bondarenko, Ortega-Cerdà, Radchenko, and Seip (BOCRS) regarding the arithmetic properties of the Hörmander–Bernhardsson extremal function $\phi$ .

The Object: $\phi$ is the unique entire function of exponential type at most $\pi$ such that $\phi(0)=1$ and $\|\phi\|_{L^1(\mathbb{R})}$ is minimal. It is the extremizer for the $L^1$ point-evaluation problem in the Paley–Wiener space $PW^1$ .
The Recurrence: BOCRS derived a three-term recurrence relation for the even Taylor coefficients of the product $f(z)f(-z)$ , where $f$ is an eigenfunction of a specific differential operator $L_{a,b}$ . The recurrence for the coefficients $u_n$ (depending on parameters $b$ and $\lambda$ ) is:
$u_{-1} = 0, \quad u_0 = 1, \quad u_{n+1} = \frac{4n+2}{n+1}\big(n(n+1)-\lambda\big)u_n + \frac{4n}{n+1}b^2 u_{n-1}$
The Conjecture: BOCRS conjectured that despite the presence of denominators ( $n+1$ ) in the recurrence, the coefficients $u_n(b, \lambda)$ are actually polynomials with integer coefficients in the variables $b^2$ and $\lambda$ . Specifically, $u_n(b, \lambda) \in \mathbb{Z}[b^2, \lambda]$ .
The Challenge: Standard holonomic arguments (D-finite series) do not explain this integrality because the underlying differential equation has irregular singularities at $0$ and $\infty$ , placing it outside the usual framework of regular-singular geometry (Picard–Fuchs equations, G-functions) where such integrality is typically found.

2. Methodology

The author proves the conjecture using a novel determinant comparison method involving Legendre polynomials. The proof proceeds in four main logical steps:

Algebraic Reduction: The author introduces a universal variable $x = b^2$ and defines a sequence $\tilde{u}_n(x, \lambda)$ . Using a lemma, the problem is reduced to proving integrality after the specialization $x = -\beta^2$ (i.e., showing $\tilde{u}_n(-\beta^2, \lambda) \in \mathbb{Z}[\beta, \lambda]$ ).
Normalization: The sequence is normalized by dividing by the central binomial coefficient $C_n = \binom{2n}{n}$ . Let $v_n = \tilde{u}_n / C_n$ .
Legendre Basis Connection: The normalized recurrence is shown to be exactly the continuant recurrence of a tridiagonal determinant arising from the action of the operator $L + \beta t - \lambda$ $L + β t - λ$ on a scaled Legendre basis $Q_n(t) = 2^n P_n(t)$ $Q_{n} (t) = 2^{n} P_{n} (t)$ , where $P_n$ $P_{n}$ are classical Legendre polynomials.
- The operator $L = -\frac{d}{dt}((1-t^2)\frac{d}{dt})$ is the Legendre differential operator.
- The recurrence coefficients match the matrix entries of this operator in the Legendre basis.
Change of Basis and Determinant Comparison:
- The same operator is represented in the standard monomial basis $\{1, t, \dots, t^n\}$ , yielding a matrix $M_n$ with integer entries (in $\mathbb{Z}[\beta, \lambda]$ ).
- The determinant of the Legendre-basis matrix is identified with the normalized sequence $v_n$ .
- By comparing the determinants of the operator in the monomial basis ( $M_n$ ) and the Legendre basis ( $N_n$ ) via a change-of-basis matrix, the author establishes the identity:
  $\det(M_n) = -C_n v_n(-\beta^2, \lambda) = -\tilde{u}_n(-\beta^2, \lambda)$
- Since $\det(M_n)$ is an integer polynomial, $\tilde{u}_n(-\beta^2, \lambda)$ must be integral. The algebraic reduction then implies the original conjecture.

3. Key Contributions and Results

A. Proof of Integrality (Theorem 1)

The paper proves that for all $n \ge 0$ :
$\tilde{u}_n(x, \lambda) \in \mathbb{Z}[x, \lambda]$
Consequently, the original coefficients satisfy $u_n(b, \lambda) \in \mathbb{Z}[b^2, \lambda]$ . This resolves Conjecture 2 of BOCRS.

B. Exact Finite-Compression Identity (Theorem 2)

The author establishes a precise link between the recurrence sequence and the determinants of finite tridiagonal matrices (BOCRS truncations). Let $T_n(a, \lambda)$ be the $n \times n$ tridiagonal matrix defined by the BOCRS spectral problem. Then:
$\tilde{u}_{n+1}(a^2, \lambda) = \binom{2n+2}{n+1} \det T_n(a, \lambda)$
This result shows that the BOCRS truncation scheme is not merely a numerical approximation but represents the exact characteristic polynomial sequence of finite principal compressions of the spectral operator, scaled by central binomial coefficients.

C. Arithmetic Consequences (Corollaries 18 & 22)

The integrality result leads to two significant arithmetic theorems regarding the Hörmander–Bernhardsson function $\phi(z) = \sum c_n z^{2n}$ :

Irrationality of Normalized Constants:
Let $C$ be the sharp point-evaluation constant for $PW^1$ , and let $\tau_n$ be the nonzero zeros of $\phi$ . Define $L_\tau(1) = \sum_{n \ge 1} \frac{(-1)^n}{\tau_n}$ .
The paper proves that the two numbers:
$\left(\frac{\pi}{4C}\right)^2 \quad \text{and} \quad -\frac{L_\tau(1)}{2C}$
cannot both be rational. This implies that at least one of the ratios $C/\pi$ or $L_\tau(1)/C$ is irrational.
Coefficient Ring Structure:
The Taylor coefficients $c_n$ of $\phi(z)$ satisfy a stronger divisibility property than previously suspected.
$c_n \in C^n \mathbb{Z}[\pi^2, C, L_\tau(1)]$
Crucially, this implies that only even powers of $\pi$ appear in the Taylor expansion of $\phi$ . Odd powers of $\pi$ are strictly forbidden due to the weighted support bound of the underlying polynomials.

D. Weighted Support Bound (Proposition 21)

The author introduces a weight function $w(x^r \lambda^s) = 2r + s$ . It is proven that for every monomial $x^r \lambda^s$ appearing in $\tilde{u}_n(x, \lambda)$ , the weight satisfies $2r + s \le n$ . This structural bound is the mechanism that forces the cancellation of denominators and restricts the powers of $\pi$ in the final coefficients.

4. Significance

Resolution of Open Problems: The paper provides the first complete proof of BOCRS Conjecture 2, complementing the author's previous proof of Conjecture 1.
New Mechanism for Integrality: Unlike typical integrality results derived from regular-singular geometry (Picard–Fuchs equations), this paper demonstrates that integrality in the presence of irregular singularities can arise from exact finite-dimensional Legendre compressions. This offers a new perspective on arithmetic properties of solutions to differential equations with irregular singularities.
Irrationality Results: The derived irrationality results for constants related to the extremal problem in Paley–Wiener spaces are non-trivial and provide new constraints on the spectral properties of the Hörmander–Bernhardsson function.
Structural Insight: The identification of the recurrence as a continuant of a tridiagonal matrix provides a concrete algebraic framework for analyzing the BOCRS spectral problem, linking orthogonal polynomials (Legendre) directly to the extremal function's coefficients.

In summary, the paper successfully bridges the gap between the analytic theory of extremal functions in harmonic analysis and deep arithmetic properties, utilizing a sophisticated determinant comparison in the Legendre basis to resolve a conjecture that resisted standard holonomic methods.