There is a question in my homeworks and I couldn’t ask to professor since she has gone abroad. I don’t have any idea abot what should I do. I really need help. I have an exam this week. Any help will be very beneficial for me. Thanks a lot
Determine two periods for pointwise limit of Fourier Series of those functions below and determine Fourier Series are uniform convergent or not without calculating Fourier coefficients.
One of functions :
$$f(x)=e^x; -1\lt x \leq 1$$
If you are expanding this function on $[-1,1]$, which means in terms of $\{ e^{in\pi x} \}_{n=-\infty}^{\infty}$, then the Fourier series converges uniformly on every iterval $[a,b]\subset(-1,1)$, but not on $[-1,1]$ because the periodic extension with period $2$ has a jump discontinuity at every integer. Uniform convergence would imply continuity.