On the Orthorecursive Expansion of Unity
Benoit Cloitre

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 in produces a sequence of rational coefficients defined by an explicit recurrence. Kalmynin and Kosenko established the bounds and through intricate -norm arguments, but left the optimal decay rates as open problems. We prove , where is the smallest real part among the zeros of a transcendental function related to the digamma function. We also improve the coefficient bound to . 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.
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Taxonomy
TopicsLinguistics, Language Diversity, and Identity · Discourse Analysis and Cultural Communication · Linguistic research and analysis
