On the Orthorecursive Expansion of Unity

1. Problem Statement

The paper investigates the asymptotic behavior of the coefficients $(c_n)_{n \ge 0}$ arising from the orthorecursive expansion of unity (the constant function $f(x)=1$) with respect to the non-orthogonal system $\{x, x^2, x^3, \dots\}$ in the Hilbert space $L^2([0, 1])$.

The coefficients are defined by the greedy algorithm of successive orthogonal projections. They satisfy the explicit recurrence relation:
$$c_0 = 1, \quad \sum_{k=0}^N c_k \frac{1}{N+k+1} = 0 \quad \text{for } N \ge 1.$$
While the recurrence is elementary, the asymptotic decay of $c_n$ and their partial sums $C_N = \sum_{k=0}^N c_k$ is highly non-trivial. Previous work by Kalmynin and Kosenko established bounds $c_n = O(n^{-3/2})$ and $C_N = O(N^{-1/2})$ but left the optimal decay rates as open problems. Kalmynin and Kosenko conjectured an oscillatory asymptotic for $c_n$ with an exponent $\delta \approx 7/3 \approx 2.33$, based on numerical evidence.

2. Methodology

The author introduces a novel analytical framework combining Volterra integral equations with Mellin transform analysis and Tauberian theorems. The methodology proceeds in four main stages:

A. Reformulation as a Volterra Equation

The discrete recurrence is recast into a continuous integral identity. By defining the partial sums $A(x) = \sum_{n \le x} c_{n-1}$ and a weighted sum $A_g(x) = \sum_{n \le x} c_{n-1} g(n/x)$ with the kernel $g(t) = \frac{2}{1+t}$, the author derives a discrete Volterra identity:
$$A_g(x) = A(x) + (A \star k)(x),$$
where $\star$ denotes multiplicative convolution and $k(y) = \frac{2y}{(1+y)^2}$.
Inverting this relation yields a Volterra integral equation of the second kind:
$$A(x) = A_g(x) - (A_g \star R)(x),$$
where $R$ is the resolvent kernel.

B. Mellin Analysis and the Resolvent

The author analyzes the problem in the Mellin domain. The modified Mellin transform of the kernel $g(t)$ is defined as:
$$g^*(z) = -z \int_0^1 g(t) t^{-z-1} dt.$$
Using properties of the digamma function $\psi$, this is shown to be:
$$g^*(z) = z \left( \psi\left(-\frac{z}{2}\right) - \psi\left(\frac{1-z}{2}\right) \right).$$
The decay of the resolvent $R(y)$ is governed by the location of the zeros of $g^*(z)$. Specifically, the Mellin transform of the resolvent is $\tilde{R}(s) = 1 - 1/g^*(-s)$.

C. Zero-Free Region Analysis

A critical component of the proof is the rigorous analytic determination of the zeros of $g^*(z)$. The author proves:

1. $g^*(z)$ has no zeros in the closed half-plane $\text{Re}(z) \le 1$.
2. There is exactly one conjugate pair of zeros in the strip $1 < \text{Re}(z) < 3/2$.
3. The zero with the smallest positive real part is $\rho_1 = \alpha_1 + i\tau_1$, where $\alpha_1 \approx 1.34652$ and $\tau_1 \approx 1.05516$.

This analysis relies on the argument principle and alternating series estimates, avoiding heavy numerical computation for the zero-free regions.

D. Tauberian Transfer and Bootstrap

Using a Volterra–Mellin transfer theorem, the author transfers the decay rate of the weighted sum $A_g(x)$ to the partial sums $A(x)$. The decay rate is determined by the real part of the rightmost zero of $g^*(z)$.
Subsequently, a bootstrap method is employed: the improved bound on partial sums is fed back into the recurrence relation via summation by parts to derive a sharper pointwise bound for the coefficients $c_n$.

3. Key Contributions and Results

Theorem 1.1 (Optimal Decay of Partial Sums)

The partial sums $C_N$ satisfy:
$$C_N = O_\varepsilon(N^{-\alpha_1 + \varepsilon})$$
for every $\varepsilon > 0$, where $\alpha_1 \approx 1.3465$ is the real part of the first non-trivial zero of $g^*(z)$. This improves upon the previous $O(N^{-1/2})$ bound.

Theorem 1.2 (Improved Pointwise Bound)

The coefficients satisfy:
$$c_n = O(n^{-2}).$$
This improves upon the previous $O(n^{-3/2})$ bound. The proof utilizes the fact that $\alpha_1 > 1$ to close the bootstrap loop.

Spectral Interpretation (Conjecture Refinement)

The paper provides a spectral interpretation of the Kalmynin–Kosenko conjecture. It suggests that the coefficients admit an asymptotic expansion dominated by the complex zeros of $g^*(z)$:
$$c_n \sim 2 \text{Re}\left( \kappa n^{-1-\rho_1} \right) = A n^{-(1+\alpha_1)} \sin(\tau_1 \log n + \phi).$$
This implies the optimal decay exponent is $\delta = 1 + \alpha_1 \approx 2.3465$, correcting the previous numerical conjecture of $\delta \approx 7/3$.

4. Significance

• Resolution of Open Problems: The paper settles the open questions regarding the optimal decay rates of the orthorecursive expansion of unity, providing rigorous proofs for bounds that were previously only conjectured or loosely estimated.
• Methodological Innovation: The development of a Volterra–Mellin transfer theorem offers a powerful new tool for analyzing discrete recurrences that resist standard summation methods (like Euler–Maclaurin or saddle-point approximations). By converting discrete structures into smooth integral equations, the author bypasses the difficulties associated with step functions and discontinuities.
• Connection to Number Theory: The problem is linked to the Nyman–Beurling approach to the Riemann Hypothesis. While the current problem is more tractable (explicit zeros), the techniques developed (Mellin analysis of resolvent kernels) may have applications in more complex arithmetic settings.
• Spectral Asymptotics: The work establishes a clear link between the arithmetic properties of the recurrence and the spectral properties (zeros) of an associated transcendental function, suggesting a trace formula analogous to those in prime number theory.

In summary, Cloitre's paper transforms a difficult discrete asymptotic problem into a solvable analytic one by leveraging the smoothness of the resolvent kernel and the precise location of the zeros of a digamma-related function, yielding sharp bounds and a refined spectral conjecture.