# periodic

• Nov 30th 2009, 02:10 PM
chihahchomahchu
periodic
I have this:
f is continuous on a closed interval [a,b] on a real line.
I defined g(x)=f(a+(b-a)x/pi) for x in closed interval [0,pi] such that f cont. on [a,b] is symmetric to function g over a the new interval [0,pi].

My question is how do I show that g is periodic?
• Nov 30th 2009, 02:37 PM
Plato
Quote:

Originally Posted by chihahchomahchu
I have this:
f is continuous on a closed interval [a,b] on a real line.
I defined g(x)=f(a+(b-a)x/pi) for x in closed interval [0,pi] such that f cont. on [a,b] is symmetric to function g over a the new interval [0,pi].
My question is how do I show that g is periodic?

Why do you think that it is periodic?

Have you actually taken examples?
Say: $a=1,~b=4,~\&~f(x)=\frac{1}{x}$
Graph that. What do you see?
• Nov 30th 2009, 03:44 PM
chihahchomahchu
Quote:

Originally Posted by Plato
Why do you think that it is periodic?

Have you actually taken examples?
Say: $a=1,~b=4,~\&~f(x)=\frac{1}{x}$
Graph that. What do you see?

Sorry for lack of information. I am derive from the Fejer-cesaro approximation theorem to prove Weierstrass approximation theorem. Fejer-cesaro theorem is periodic 2pi. In order for me to use Fejer-cesaro theorem, I need to do a change of variable of function f cont. on [a,b] to new function and new interval, so I can define the Fourier nth partial sum.

How do I make this new function periodic?
• Nov 30th 2009, 06:13 PM
chihahchomahchu
in other word, let f(-x)=f(x), x in [0,pi], How do I take the unique periodic extension of this function?