Hartogs–Rosenthal theorem

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.

Contenuti 1 Dichiarazione 2 Prova 3 Guarda anche 4 Appunti 5 References 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 {stile di visualizzazione con} e {stile di visualizzazione {sopra {z}}} .

So it suffices to show that {stile di visualizzazione {sopra {z}}} can be uniformly approximated by a rational function on K.

Sia g(z) be a smooth function of compact support on C equal to 1 on K and set {stile di visualizzazione f(z)=g(z)cdot {sopra {z}}.} By the generalized Cauchy integral formula {stile di visualizzazione f(z)={frac {1}{2pi io}}iint _{Cbackslash K}{frac {parziale f}{parziale {sbarra {w}}}}{frac {dwwedge d{sbarra {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 {stile di visualizzazione {sbarra {z}}} by a rational function.[1] See also Runge's theorem Mergelyan's theorem Notes ^ Conway 2000 References Conway, Giovanni B. (1995), Functions of one complex variable II, Testi di laurea in Matematica, vol. 159, Springer, p. 197, ISBN 0387944605 Conway, Giovanni B. (2000), A course in operator theory, Laurea Magistrale in Matematica, vol. 21, Società matematica americana, pp. 175–176, ISBN 0821820656 Gamelin, Theodore W. (2005), Uniform algebras (2nd ed.), Società matematica americana, pp. 46–47, ISBN 0821840495 Hartogs, Friedrichs; Rosenthal, Artù (1931), "Über Folgen analytischer Funktionen", Annali matematici, 104: 606–610, doi:10.1007/bf01457959, S2CID 179177370 Categorie: Rational functionsTheorems in approximation theoryTheorems in complex analysis