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.
Proofedit
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
Restricting z to K and taking Riemann approximating sums for the integral on the right hand side yields the required uniform approximation of by a rational function.[1]
hartogs, rosenthal, theorem, mathematics, classical, result, complex, analysis, uniform, approximation, continuous, functions, compact, subsets, complex, plane, rational, functions, theorem, proved, 1931, german, mathematicians, friedrich, hartogs, arthur, ros. 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 Contents 1 Statement 2 Proof 3 See also 4 Notes 5 ReferencesStatement editThe 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 editBy the Stone Weierstrass theorem any complex valued continuous function on K can be uniformly approximated by a polynomial in z displaystyle z nbsp and z displaystyle overline z nbsp So it suffices to show that z displaystyle overline z nbsp 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 f z g z z displaystyle f z g z cdot overline z nbsp By the generalized Cauchy integral formula f z 12pi C K f w dw dw w z 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 nbsp 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 z displaystyle bar z nbsp by a rational function 1 See also editRunge s theorem Mergelyan s theoremNotes edit Conway 2000References editConway 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 Uber Folgen analytischer Funktionen Mathematische Annalen 104 606 610 doi 10 1007 bf01457959 S2CID 179177370 Retrieved from https en wikipedia org w index php title Hartogs Rosenthal theorem amp oldid 1013816753, wikipedia, wiki, book, books, library,
