Factorization of solutions of linear differential equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929681576493056 |
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| author | Gröhn, Janne |
| author_facet | Gröhn, Janne |
| contents | 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 $\mathbb{C}$. 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 $g\in{\rm BMOA}$ satisfies the interpolation property $$g'(z_n) = -\frac{1}{2} \, \frac{B''(z_n)}{B'(z_n)}, \quad z_n\inΛ.$$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_09508 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Factorization of solutions of linear differential equations Gröhn, Janne Classical Analysis and ODEs Complex Variables Primary 34M10, Secondary 30H30, 30H35 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 $\mathbb{C}$. 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 $g\in{\rm BMOA}$ satisfies the interpolation property $$g'(z_n) = -\frac{1}{2} \, \frac{B''(z_n)}{B'(z_n)}, \quad z_n\inΛ.$$ 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. |
| title | Factorization of solutions of linear differential equations |
| topic | Classical Analysis and ODEs Complex Variables Primary 34M10, Secondary 30H30, 30H35 |
| url | https://arxiv.org/abs/2501.09508 |