Rational functions as new variables

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A1 Alkuperäisartikkeli tieteellisessä aikakauslehdessä
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Authors

Date

2022-07

Department

Department of Mathematics and Systems Analysis

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Mcode

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Language

en

Pages

22
1-22

Series

Banach Journal of Mathematical Analysis, Volume 16, issue 3

Abstract

In multicentric calculus, one takes a polynomial p with distinct roots as a new variable and represents complex valued functions by Cd-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) | ≤ ρ. 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) | ≤ ρ 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.

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Publisher Copyright: © 2022, The Author(s).

Keywords

Functional calculus, Rational functions, Series expansions

Other note

DOI

10.1007/s43037-022-00189-3

Citation

Andrei, D, Nevanlinna, O & Vesanen, T 2022, ' Rational functions as new variables ', Banach Journal of Mathematical Analysis, vol. 16, no. 3, 37, pp. 1-22 . https://doi.org/10.1007/s43037-022-00189-3

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https://urn.fi/URN:NBN:fi:aalto-202205243386

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[article-cris] Perustieteiden korkeakoulu / SCI