Analytic approximation of matrix functions in $L^p$
| dc.creator | Baratchart, L. | |
| dc.creator | Nazarov, F. L. | |
| dc.creator | Peller, V. V. | |
| dc.date | 2008-05-28 | |
| dc.date.accessioned | 2026-07-07T09:41:22Z | |
| dc.date.available | 2026-07-07T09:41:22Z | |
| dc.description | We consider the problem of approximation of matrix functions of class $L^p$ on the unit circle by matrix functions analytic in the unit disk in the norm of $L^p$, $2\le p<\be$. For an $m\times n$ matrix function $Φ$ in $L^p$, we consider the Hankel operator $H_Φ:H^q(C^n)\to H^2_-(C^m)$, $1/p+1/q=1/2$. It turns out that the space of $m\times n$ matrix functions in $L^p$ splits into two subclasses: the set of respectable matrix functions and the set of weird matrix functions. If $Φ$ is respectable, then its distance to the set of analytic matrix functions is equal to the norm of $H_Φ$. For weird matrix functions, to obtain the distance formula, we consider Hankel operators defined on spaces of matrix functions. We also describe the set of $p$-badly approximable matrix functions in terms of special factorizations and give a parametrization formula for all best analytic approximants in the norm of $L^p$. Finally, we introduce the notion of $p$-superoptimal approximation and prove the uniqueness of a $p$-superoptimal approximant for rational matrix functions. | |
| dc.description | 43 pages | |
| dc.identifier | https://arxiv.org/abs/0805.4366 | |
| dc.identifier | http://arxiv.org/abs/0805.4366 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/161804 | |
| dc.subject | Functional Analysis | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Complex Variables | |
| dc.subject | 47B35; 30D55; 30E10 | |
| dc.title | Analytic approximation of matrix functions in $L^p$ | |
| dc.type | text |