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Math Help - Need help with connection of theorems

  1. #1
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    Need help with connection of theorems

    I need help with using/derive fejer-cesaro approximation to prove weierstrass approximation theorem: every continuous function on [a,b] can be approximate uniformly by polynomials.

    How do I find a continuous function g on an interval of fejer-cesaro that is isomorphic to f:[a,b] in weierstrass approximation theorem?

    The fejer-cesaro given: An(x)=(So+S2+S3+.......+S(n-1))/N with the fourier series nth partial sum Sf(x)=Summation of Cke^(jkx) where k=-n to n and x is in [0,2pi]

    and I need help to prove An(x) is a polynomial.

    I would be very appreciate if someone can give me hints of find g and show that An(x) is a polynomial.
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  2. #2
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    Quote Originally Posted by chihahchomahchu View Post
    I need help with using/derive fejer-cesaro approximation to prove weierstrass approximation theorem: every continuous function on [a,b] can be approximate uniformly by polynomials.

    How do I find a continuous function g on an interval of fejer-cesaro that is isomorphic to f:[a,b] in weierstrass approximation theorem?

    The fejer-cesaro given: An(x)=(So+S2+S3+.......+S(n-1))/N with the fourier series nth partial sum Sf(x)=Summation of Cke^(jkx) where k=-n to n and x is in [0,2pi]

    and I need help to prove An(x) is a polynomial.

    I would be very appreciate if someone can give me hints of find g and show that An(x) is a polynomial.
    A_n(x) is not a polynomial, but it is a finite linear combination of exponential functions e^{ikx}. You can approximate each of these exponentials uniformly on the interval [a,b] by taking sufficiently many terms in its power series expansion.

    So you get the Weierstrass result in two stages. First, use the Fejér–Cesŕro result to approximate f uniformly by the function A_n(x). Then use the power series for the exponential function to approximate A_n(x) uniformly by a polynomial.
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  3. #3
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    Quote Originally Posted by Opalg View Post
    A_n(x) is not a polynomial, but it is a finite linear combination of exponential functions e^{ikx}. You can approximate each of these exponentials uniformly on the interval [a,b] by taking sufficiently many terms in its power series expansion.

    So you get the Weierstrass result in two stages. First, use the Fejér–Cesŕro result to approximate f uniformly by the function A_n(x). Then use the power series for the exponential function to approximate A_n(x) uniformly by a polynomial.
    Thanks so much for the reply,

    So I can do a change of variable as g(x)=f(a+(b-a)x/pi), for x in [0,pi] then g cont. on [0,pi] and use the fejer-cesaro approximation to prove that An(x) is uniformly continuous on [0,pi]. Then I am left to prove that An(x) is a polynomial. Is this exactly the outline of the proof or I am missing something else?
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  4. #4
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    Quote Originally Posted by chihahchomahchu View Post
    Then I am left to prove that An(x) is a polynomial.
    No. As I already said, A_n(x) is not a polynomial. But it can be uniformly approximated by a polynomial, using the fact that e^{ikx} = 1+ikx +\frac{(ikx)^2}{2!} + \frac{(ikx)^3}{3!}+\ldots. By taking sufficiently many terms in that series, you can get a polynomial that is uniformly close to e^{ikx} (on any finite interval [a,b]).
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  5. #5
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    Quote Originally Posted by Opalg View Post
    No. As I already said, A_n(x) is not a polynomial. But it can be uniformly approximated by a polynomial, using the fact that e^{ikx} = 1+ikx +\frac{(ikx)^2}{2!} + \frac{(ikx)^3}{3!}+\ldots. By taking sufficiently many terms in that series, you can get a polynomial that is uniformly close to e^{ikx} (on any finite interval [a,b]).

    Thanks
    How do you explain the connection between weierstrass theorem and fejer-cesaro theorem where weierstrass theorem is base on the whole real line and fejer-cesaro is based on a circle.
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