On the Orthorecursive Expansion of Unity

Abstract

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 $c_n=O(n^{-3/2})$ 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 $C_N=O_{\epsilon }(N^{-{\alpha }_1+\epsilon })$, 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.

arXiv comment12 pages. Complete rewrite with new title. The main results are unchanged, but the method of proof is entirely new. The Perron-based approach of v1 is replaced by a Volterra-Mellin transfer theorem. A new pointwise bound $c_n=O(n^{-2})$ is obtained by a bootstrap argument. The spectral analysis of the zeros is simplified. MSC classes updated