[Paper Review] Mean asymptotic behaviour of radix-rational sequences and dilation equations (Extended version)

Philippe Dumas|ArXiv.org|Jul 9, 2008

Advanced Numerical Analysis Techniques37 references3 citations

TL;DR

This paper establishes a general asymptotic expansion for the Cesàro means of radix-rational sequences using linear algebra and dilation equations. It shows that the mean behavior is governed by periodic functions and eigenvalues derived from the joint spectral radius, with precise error bounds based on the spectral properties of the underlying linear representation.

ABSTRACT

The generating series of a radix-rational sequence is a rational formal power series from formal language theory viewed through a fixed radix numeration system. For each radix-rational sequence with complex values we provide an asymptotic expansion for the sequence of its Cesàro means. The precision of the asymptotic expansion depends on the joint spectral radius of the linear representation of the sequence; the coefficients are obtained through some dilation equations. The proofs are based on elementary linear algebra.

Motivation & Objective

• To develop a general framework for analyzing the mean asymptotic behavior of radix-rational sequences.
• To extend classical asymptotic methods for rational sequences to the more complex setting of radix-rational sequences.
• To provide a unified approach using linear algebra and dilation equations for sequences defined by scaling transformations in a fixed radix system.
• To characterize the structure of the asymptotic expansion in terms of periodic functions and eigenvalues derived from the linear representation.

Proposed method

• The paper models radix-rational sequences via a linear representation using matrices of size d, where d is the dimension of the state space.
• It introduces a dilation equation for vector-valued functions that encode the self-similar structure of the sequence under scaling by the radix B.
• The asymptotic expansion is derived by analyzing the solution of the dilation equation using the joint spectral radius of the transformation matrices.
• The method employs Jordan decomposition and noise analysis to handle the spectral decomposition of the linear system.
• It uses periodic functions and complex roots of unity to describe oscillatory components in the asymptotic scale.
• The approach leverages conjugate symmetry and rotation dynamics in the complex plane to describe the long-term behavior of the Cesàro means.

Experimental results

Research questions

• RQ1 How can the mean asymptotic behavior of a radix-rational sequence be systematically analyzed beyond ad hoc methods?
• RQ2 What role does the joint spectral radius play in determining the growth rate and oscillatory components of the Cesàro mean?
• RQ3 In what way do dilation equations capture the self-similar structure of radix-rational sequences?
• RQ4 Why do periodic and pseudo-periodic functions emerge naturally in the asymptotic expansion of such sequences?
• RQ5 Under what conditions is the asymptotic expansion periodic or almost periodic?

Key findings

• The Cesàro mean of any radix-rational sequence admits an asymptotic expansion of the form $ \sum_{\alpha > \alpha_*, \ell \geq 0} N^\alpha \log_B^\ell(N) \sum_\omega \omega^{\lfloor \log_B N \rfloor} \Psi_{\alpha,\ell,\omega}(\log_B N) + O(N^{\alpha_*}) $, where $ \Psi $ are 1-periodic functions.
• The coefficients in the expansion are determined by solving a system of dilation equations for vector-valued functions with specific boundary conditions.
• When the angle $ \vartheta $ in the spectral decomposition is commensurate with $ \pi $, the asymptotic arc $ \mathbf{\Gamma}(t) $ becomes periodic with period $ 2q $, where $ \vartheta/\pi = p/q $ in lowest terms.
• For irrational $ \vartheta/\pi $, the asymptotic arc is non-periodic but invariant under rotation by $ \vartheta $, leading to quasiperiodic behavior.
• The solution to the dilation equation is unique and satisfies a symmetry condition $ P\mathbf{\Phi} = \mathbf{\Phi} $, which enforces a specific geometric constraint on the vector-valued function.
• The asymptotic behavior is governed by a rotation about a fixed point $ \Omega $, with the vector $ \mathbf{\Gamma}(t+1) $ obtained by rotating $ \mathbf{\Gamma}(t) $ by angle $ \vartheta $.