Results 1 to 2 of 2

Math Help - Stone-Weiersrass thm.-related questions

  1. #1
    Newbie
    Joined
    Jan 2010
    Posts
    14

    Stone-Weiersrass thm.-related questions

    (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).
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by matzerath View Post
    (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 \bold{x}\ne\bold{y} then we can define a map from X\to\mathbb{R} by sending an arbitrary ordered pair to the difference between the kth coordinate of it and the kth coordinate of \bold{x}, thus if f is that function we have that 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 \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?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. stone problem
    Posted in the Calculus Forum
    Replies: 1
    Last Post: February 27th 2010, 04:16 PM
  2. Stone throwing velocity problem
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 20th 2009, 07:03 PM
  3. Stone-throwing problem
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: January 7th 2009, 08:29 AM
  4. a falling stone with sound
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: September 2nd 2007, 06:24 PM
  5. Falling stone!
    Posted in the Math Topics Forum
    Replies: 2
    Last Post: July 18th 2006, 06:28 AM

Search Tags


/mathhelpforum @mathhelpforum