Factorization of solutions of linear differential equations

2025/01/16 by Janne Gröhn, Gröhn, Janne
Computer Science · Mathematics · #30H35 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Differential Equations and Numerical Methods #FOS: Mathematics #Matrix Theory and Algorithms #Numerical methods for differential equations #Primary 34M10 #Secondary 30H30

paper · pdf · doi:10.48550/arxiv.2501.09508

openalex publication_date 2025/01/16 · openalex created_date 2025/01/18 · openalex updated_date 2026/08/03

Abstract

This paper supplements recents results on linear differential equations f''+Af=0, where the coefficient A is analytic in the unit disc of the complex plane ℂ. It is shown that, if A is analytic and |A(z)|²(1-|z|²)³ dm(z) is a Carleson measure, then all non-trivial solutions of f''+Af=0 can be factorized as f=Be^g, where B is a Blaschke product whose zero-sequence Λ is uniformly separated and where g∈\rm BMOA satisfies the interpolation property g'(z_n) = -(1)/(2) (B''(z_n))/(B'(z_n)), z_n∈Λ. Among other things, this factorization implies that all solutions of f''+Af=0 are functions in a Hardy space and have no singular inner factors. Zero-free solutions play an important role as their maximal growth is similar to the general case. The study of zero-free solutions produces a new result on Riccati differential equations.

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