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On the possibility of division and involution to a fractional power in the algebra of rational functions
P. V. Paramonov
Suppose that a function $f(z)$ satisfies a Lipschitz condition with an arbitrary positive element on a compact set $X$ in $\mathbf C$ and can be uniformly approximated on $X$ by rational functions. If $q>1$ and some branch of $(f(z))^q$ is continuous on $X$, then this branch can also be approximated on $X$ by rational functions. Also, an example is given of a compact set $X$ and two functions $f(z)$ and $g(z)$ uniformly approximable on $X$ by rational functions and with ratio $g(z)/f(z)$ naturally (uniquely) defined and continuous on $X$ but not approximable by rational functions.
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Mathematics of the USSR-Izvestiya, 1988, 30:2, 385–393
MSC: Primary 41A20, 46J10; Secondary 30C15, 41A46
P. V. Paramonov, “On the possibility of division and involution to a fractional power in the algebra of rational functions”, Izv. Akad. Nauk SSSR Ser. Mat., 51:2 (1987), 412–420; Math. USSR-Izv., 30:2 (1988), 385–393
Citation in format AMSBIB
\paper On~the possibility of division and involution to a~fractional power in the algebra of rational functions
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\jour Math. USSR-Izv.
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