Question

Why does fourier series doesnt work when evaluating at discontinuities?

Answer 1

Answer:

I was reading about Fourier series and have a doubt concerning it. The book I am reading from does not seem to help. As I understand, {e0=12√,e1=sin(x),e2=cos(x),e3=sin(2x),e4=cos(2x)⋯} is a basis for the inner product space of piecewise continuous functions in [−π,π] with inner product <f,g>=1π∫π−πfg¯. Hence any function in this space may be represented by f=∑∞k=0<f,ek>ek. My question is what happens at points of discontinuity x. As f is identical with the series (which by the way is unclear to me as to why it coverges) shouldn't f(x) be identical with the series at x, i.e. f(x)=∑∞k=0<f,ek>ek(x). But Dirichlet's theorem (stated without proof in my book) says that at points of discontinuity, the series ∑∞k=0<f,ek>ek(x) converges to f(x−)+f(x+)2 and not to f(x). Why is this so? Thanks.