Fourier Series

[Dyls]

I'm a little confused about the difference between the half range fourier series and the full range fourier series. What is the difference between the two in an odd function like f(x)=x and an even function like f(x)=x^2 ? Maybe an example to clear things up. Thank you.

[matt grime]

Given a function from [0,1] to R where f(0)=0, there are several ways to make from it a periodic function on some interval. The simplest is to repeat the function by setting f(x+1)=f(x). However, there is another option, and one that cuts your work in half.

We extend the function to [-1.1] by

1) refelcting in the y axis, ie set f(-x) = f(x)

2) rotate 180 degrees about the origin by setting f(-x)=-f(x)

then repeat these to get a periodic function.

The first is even on the interval [-1,1] so it only has cosines in its fourier series, the second is odd so only has sines.

So, take f(x)=x on [0,1] if we extend it to an even funciton on [-1,1] then we get |x|, if we extend to an odd function we just get x. The first has a fourier series only using cosines, the second only using sines.