A Tauberian approach to the orthorecursive expansion of unity

We establish a Tauberian theorem connecting the unknown asymptotic behavior of the partial sums ∑n≤ xaₙ to the known asymptotics of weighted sums ∑n≤ xaₙg(n/x), as x→∞, where g:(0,1] is a given function. Our approach relies on an identity relating a modified Mellin transform of g to the Dirichlet series ∑n≥1aₙn⁻ˢ. As an application, we solve an open problem posed by Kalmynin and Kosenko regarding the "orthorecursive expansion of unity" associated with a sequence (cₙ)n≥0. Specifically, we improve their partial-sum bound CN=∑0≤ n≤ Ncₙ=O(N-1/2), by obtaining the optimal estimate CN=O(N^-α₁+ε), where α₁≈1.3465165 is the smallest real part among the zeros of a transcendental function related to the digamma function.