On the Orthorecursive Expansion of Unity

TL;DR

This paper improves the understanding of the decay rates of coefficients in the orthorecursive expansion of unity, establishing sharper bounds and revealing connections to zeros of a transcendental function.

Contribution

It proves sharper bounds for the coefficients and partial sums in the orthorecursive expansion, using a novel Tauberian approach involving Volterra integral equations.

Findings

Coefficient decay rate improved to O(n^{-2})

Partial sum bounds established as O(N^{- ext{approx.}1.3465})

Method employs Mellin analysis and contour shifting techniques

Abstract

The orthorecursive expansion of unity with respect to the system $\{x, x^2, x^3, \ldots\}$ in $L^2([0,1])$ produces a sequence of rational coefficients $(c_n)$ defined by an explicit recurrence. Kalmynin and Kosenko established the bounds $c_n = O(n^{-3/2})$ and $C_N = \sum_{k=0}^{N} c_k = O(N^{-1/2})$ through intricate $L^2$-norm arguments, but left the optimal decay rates as open problems. We prove $C_N = O_{\varepsilon}(N^{-\alpha_1+\varepsilon})$, where $\alpha_1 \approx 1.3465$ is the smallest real part among the zeros of a transcendental function related to the digamma function. We also improve the coefficient bound to $c_n = O(n^{-2})$. The method rests on a Tauberian transfer theorem that recasts the discrete recurrence as a Volterra integral equation, whose resolvent is smooth and amenable to Mellin analysis and contour shifting.