# Thread: Need help with connection of theorems

1. ## 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.

2. Originally Posted by chihahchomahchu
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.

3. Originally Posted by Opalg
$A_n(x)$ 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?

4. Originally Posted by chihahchomahchu
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]).

5. Originally Posted by Opalg
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.