Rational functions as new variables

Abstract In multicentric calculus , one takes a polynomial p with distinct roots as new variable and represents complex valued functions by $${\mathbb {C}}^d$$ C d -valued functions, where d is the degree of . An application e.g. possibility to represent piecewise constant holomorphic function convergent power series, simultaneously in all components $$|p(z)| \le \rho$$ | p ( z ) ≤ ρ this paper, we study necessary modifications needed, if take rational $$r=p/q$$ r = / q instead. This allows consider defined neighborhoods any compact set opposed case domains are always polynomially convex. Two applications formulated. One giving series expression for Sylvester equations $$AX-XB =C$$ A X - B general A B being bounded operators Banach spaces spectra. The other formulates K-spectral result Hilbert spaces.