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Rational functions as new variables
- Publication Year :
- 2021
-
Abstract
- In multicentric calculus one takes a polynomial $p$ with distinct roots as a new variable and represents complex valued functions by $\mathbb C^d$-valued functions, where $d$ is the degree of $p$. An application is e.g. the possibility to represent a piecewise constant holomorphic function as a convergent power series, simultaneously in all components of $|p(z)| \le \rho$. In this paper we study the necessary modifications needed, if we take a rational function $r=p/q$ as the new variable instead. This allows to consider functions defined in neighborhoods of any compact set as opposed to the polynomial case where the domains $|p(z)| \le \rho$ are always polynomially convex. Two applications are formulated. One giving a convergent power series expression for Sylvester equations $AX-XB =C$ in the general case of $A,B$ being bounded operators in Banach spaces with distinct spectra. The other application formulates a K-spectral result for bounded operators in Hilbert spaces.<br />Comment: 20 pages, 5 figures
- Subjects :
- Mathematics - Complex Variables
30B10, 30C10, 30E99, 46J10, 47A25, 47A60
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2104.11088
- Document Type :
- Working Paper