Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Tauberian approach to the orthorecursive expansion of unity

Benoit Cloitre

TL;DR

This paper establishes a Tauberian framework that links the unknown asymptotics of unweighted partial sums $A(x)=\sum_{n\le x} a_n$ to known asymptotics of weighted sums $A_g(x)=\sum_{n\le x} a_n g(n/x)$ via a modified Mellin transform $g^{\star}$ and its zeros. The key device is the identity $L(a,s)=s\frac{A_g^{*}(-s)}{g^{\star}(-s)}$, enabling a Perron-type contour analysis to translate weighted behavior into unweighted asymptotics, with the asymptotics governed by $\alpha_g$ (the leftmost zero of $g^{\star}$) and the main term parameters. The authors apply this to the orthorecursive expansion of unity studied by Kalmynin and Kosenko, transforming the recurrence into a weighted-sum setting with $g(t)=\frac{2}{1+t}$ and showing $A_g(x)=\mathcal{O}(x^{-3/2})$. They identify the principal zero of $g^{\star}$ as $\alpha_1=\operatorname{Re}(\rho_1)\approx1.3465$, and deduce the optimal decay $C_N=\sum_{n\le N} c_n=\mathcal{O}(N^{-\alpha_1+\varepsilon})$ for any $\varepsilon>0$, thereby resolving the open problem. The results highlight a versatile Tauberian method that combines Mellin–Flajolet techniques with zero analysis of digamma-structured transforms, with potential applicability to other $L^2$-approximation problems and related Tauberian questions.

Abstract

We establish a Tauberian theorem connecting the unknown asymptotic behavior of the partial sums $\sum_{n\le x}a_{n}$ to the known asymptotics of weighted sums $\sum_{n\le x}a_{n}g(n/x)$, as $x\rightarrow\infty$, where $g:(0,1]\to\mathbb{R}$ is a given function. Our approach relies on an identity relating a modified Mellin transform of $g$ to the Dirichlet series $\sum_{n\ge1}a_{n}n^{-s}$. As an application, we solve an open problem posed by Kalmynin and Kosenko regarding the "orthorecursive expansion of unity" associated with a sequence $(c_{n})_{n\geq0}$. Specifically, we improve their partial-sum bound $C_{N}=\sum_{0\leq n\leq N}c_{n}=\mathcal{O}(N^{-1/2})$, by obtaining the optimal estimate $C_{N}=\mathcal{O}(N^{-α_{1}+ε})$, where $α_{1}\approx1.3465165$ is the smallest real part among the zeros of a transcendental function related to the digamma function.

A Tauberian approach to the orthorecursive expansion of unity

TL;DR

This paper establishes a Tauberian framework that links the unknown asymptotics of unweighted partial sums to known asymptotics of weighted sums via a modified Mellin transform and its zeros. The key device is the identity , enabling a Perron-type contour analysis to translate weighted behavior into unweighted asymptotics, with the asymptotics governed by (the leftmost zero of ) and the main term parameters. The authors apply this to the orthorecursive expansion of unity studied by Kalmynin and Kosenko, transforming the recurrence into a weighted-sum setting with and showing . They identify the principal zero of as , and deduce the optimal decay for any , thereby resolving the open problem. The results highlight a versatile Tauberian method that combines Mellin–Flajolet techniques with zero analysis of digamma-structured transforms, with potential applicability to other -approximation problems and related Tauberian questions.

Abstract

We establish a Tauberian theorem connecting the unknown asymptotic behavior of the partial sums to the known asymptotics of weighted sums , as , where is a given function. Our approach relies on an identity relating a modified Mellin transform of to the Dirichlet series . As an application, we solve an open problem posed by Kalmynin and Kosenko regarding the "orthorecursive expansion of unity" associated with a sequence . Specifically, we improve their partial-sum bound , by obtaining the optimal estimate , where is the smallest real part among the zeros of a transcendental function related to the digamma function.
Paper Structure (16 sections, 6 theorems, 67 equations)

This paper contains 16 sections, 6 theorems, 67 equations.

Table of Contents

  1. The Tauberian Theorem
  2. The Mellin-Flajolet trick
  3. Application to the Orthorecursive Expansion
  4. Transformation of the Problem
  5. The Mellin transform of g
  6. Zero-free region for $g^{\star}$
  7. Location of the principal zero
  8. Case 1: $\operatorname{Re}(z)\in[-0.25,0]$
  9. Case 2: $\operatorname{Re}(z)\in(0,2.5]$
  10. Analysis and Bound for the Main Part $T_{\text{main}}(z)$
  11. Bound for the Remainder Part $T_{\text{rem}}(z)$
  12. Conclusion for $|g^{\star}(z)-1|$ with $\operatorname{Re}(z)\in(0,2.5]$ and $y_{z}\ge10$
  13. Overall Conclusion for $D_{10}$
  14. Proof of Theorem \ref{['thm:ortho_main_app_restored_6']}
  15. PARI/GP Computation for Argument Principle
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $g:(0,1]\to\mathbb{R}$ be a function such that its modified Mellin transform exists for $\operatorname{Re}(z)<0$ and can be analytically continued. Assume further that: Let $(a_{n})_{n\ge1}$ be a sequence of real numbers. Define the partial sum $A(x)=\sum_{n\le x}a_{n}$ and the weighted sum $A_{g}(x)=\sum_{n\le x}a_{n}g(n/x)$. Suppose that for some fixed real values $\beta$ and $c$: (If $\b

Theorems & Definitions (12)

  • Theorem 1.1: Tauberian Theorem
  • proof
  • Theorem 2.1: Orthorecursive theorem
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 2 more