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 of theorem
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 of theorem
By the Stone–Weierstrass theorem any complex-valued continuous function on K can be uniformly approximated by a polynomial in and .
So it suffices to show that 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
since K has measure zero.
- Conway, John B. (1995), Functions of one complex variable II, Graduate Texts in Mathematics, 159, Springer, p. 197, ISBN 0387944605
- Conway, John B. (2000), A course in operator theory, Graduate Studies in Mathematics, 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