Hartogs–Rosenthal theorem

In mathematics, the Hartogs–Rosenthal theorem is a classical result in complex analysis on the uniform approximation of continuous functions on compact subsets of the complex plane by rational functions. The theorem was proved in 1931 by the German mathematicians Friedrich Hartogs and Arthur Rosenthal and has been widely applied, particularly in operator theory.

Statement

The Hartogs–Rosenthal theorem states that if K is a compact subset of the complex plane with Lebesgue measure zero, then any continuous complex-valued function on K can be uniformly approximated by rational functions.

Proof

By the Stone–Weierstrass theorem any complex-valued continuous function on K can be uniformly approximated by a polynomial in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} and ${\overline {z}}$.

So it suffices to show that ${\overline {z}}$ can be uniformly approximated by a rational function on K.

Let g(z) be a smooth function of compact support on C equal to 1 on K and set

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z)=g(z)\cdot \overline{z}.}

By the generalized Cauchy integral formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z) = \frac{1}{2\pi i}\iint_{{C\backslash K}} \frac{\partial f}{\partial \bar{w}}\frac{dw\wedge d\bar{w}}{w-z}, }

since K has measure zero.

Restricting z to K and taking Riemann approximating sums for the integral on the right hand side yields the required uniform approximation of ${\bar {z}}$ by a rational function.

See also

• Runge's theorem
• Mergelyan's theorem

References

• Conway, John B. (1995), Functions of one complex variable II, Graduate Texts in Mathematics, vol. 159, Springer, p. 197, ISBN 0387944605
• Conway, John B. (2000), A course in operator theory, Graduate Studies in Mathematics, vol. 21, American Mathematical Society, pp. 175–176, ISBN 0821820656
• Gamelin, Theodore W. (2005), Uniform algebras (2nd ed.), American Mathematical Society, pp. 46–47, ISBN 0821840495
• Hartogs, Friedrichs; Rosenthal, Arthur (1931), "Über Folgen analytischer Funktionen", Mathematische Annalen, 104: 606–610, doi:10.1007/bf01457959, S2CID 179177370