Would you expect to see the Gibbs phenomenon in the plot of

Would you expect to see the Gibb\'s phenomenon in the plot of the Fourier Series for f(x)? If so, for what values of x and why?

Solution

The Gibbs Phenomena occurs when there is a sudden jump or discontinuity in the function at some point, the only point where it can happen is the point x=1, so we need to check the left limit and right limit at these points

f(1-h) = 1 - 1^2 = 0

f(1+h) = sin(pi)sin(pi/3) = 0

Since the function is continuous at x=1, hence there will be no point of discontinuity in the funtion and no gibbs phenomena will be going to occur