On the Orthorecursive Expansion of Unity

The orthorecursive expansion of unity with respect to the system $\{x,x^2,x^3,\dots \}$ in $L^2([0,1])$ produces a sequence of rational coefficients $(c_n)$ defined by an explicit recurrence. Kalmynin and Kosenko established the bounds $c_n=O(n^{-3\mathrm{/}2})$ and $C_N={\sum }_{k=0}^N{c}_k=O(N^{-1\mathrm{/}2})$ through intricate $L^2$-norm arguments, but left the optimal decay rates as open problems. We prove $C_N=O_{\epsilon }(N^{-{\alpha }_1+\epsilon })$, 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.