Stone-Weiersrass thm.-related questions

Jan 2010
14
1
(1) If f(x,y) is a real function and continuous on the closed rectangle X=[a,b]x[c,d] in the Euclidean plane R^n.
Show that f can be uniformly approximated on X by polynomials in x ad in y with real coefficients.


(2) Let X be the closed unit disc in the complex plane (denoted C).
Show that any function in C(X,C) can be uniformly approx on X by polynomials in z and conjugate z with complex coefficients.


(3) Let X and Y be compact Hausdorff spaces, and f a function in C(XxY,C).
Show that f can be uniformly approx. by functions of the form Sum(from 1 to n)(f_i)(g_i), where the f_i's are in C(X,C) and the g_i's are in C(Y,C).
 

Drexel28

MHF Hall of Honor
Nov 2009
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Berkeley, California
(1) If f(x,y) is a real function and continuous on the closed rectangle X=[a,b]x[c,d] in the Euclidean plane R^n.
Show that f can be uniformly approximated on X by polynomials in x ad in y with real coefficients.
Clearly if \(\displaystyle \bold{x}\ne\bold{y}\) then we can define a map from \(\displaystyle X\to\mathbb{R}\) by sending an arbitrary ordered pair to the difference between the kth coordinate of it and the kth coordinate of \(\displaystyle \bold{x}\), thus if \(\displaystyle f\) is that function we have that \(\displaystyle f(\bold{x})=0,f(\bold{y})=0\). Thus, the set of all polynomials separates points and since clearly there are infinitely many constant functions in your space it follows from the S.W.T. that the closure of the set of all polynomials is \(\displaystyle \mathcal{C}[X]\). Conclude

(2) Let X be the closed unit disc in the complex plane (denoted C).
Show that any function in C(X,C) can be uniformly approx on X by polynomials in z and conjugate z with complex coefficients.
Similar, let's see some work (you will clearly need to look at the complex version of the S.W.T)

(3) Let X and Y be compact Hausdorff spaces, and f a function in C(XxY,C).
Show that f can be uniformly approx. by functions of the form Sum(from 1 to n)(f_i)(g_i), where the f_i's are in C(X,C) and the g_i's are in C(Y,C).
Slightly trickier, ideas?