In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Hartogs–Rosenthal theorem is a classical result in
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...
on the
uniform approximation of continuous functions on compact subsets of the
complex plane
In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
by
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...
s. The theorem was proved in 1931 by the German mathematicians
Friedrich Hartogs and
Arthur Rosenthal and has been widely applied, particularly in
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operato ...
.
Statement
The Hartogs–Rosenthal theorem states that if ''K'' is a compact subset of the complex plane with
Lebesgue measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
zero, then any continuous complex-valued function on ''K'' can be uniformly approximated by rational functions.
Proof
By the
Stone–Weierstrass theorem
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval (mathematics), interval can be uniform convergence, uniformly approximated as closely as desired by a polynomial fun ...
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
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain.
A function of class C^k is a function of smoothness at least ; t ...
of compact support on C equal to 1 on ''K'' and set
:
By the
generalized Cauchy integral formula
:
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
by a rational function.
See also
*
Runge's theorem
*
Mergelyan's theorem Mergelyan's theorem is a result from approximation by polynomials in complex analysis proved by the Armenian mathematician Sergei Mergelyan in 1951.
Statement
:Let K be a compact subset of the complex plane \mathbb C such that \mathbb C \setm ...
Notes
References
*
*
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{{DEFAULTSORT:Hartogs-Rosenthal theorem
Rational functions
Theorems in approximation theory
Theorems in complex analysis