# Uniqueness of fourier series

1. Nov 17, 2009

### boombaby

1. The problem statement, all variables and given/known data
Suppose that f is an integrable function (and suppose it's real valued) on the circle with c_n=0 for all n, where c_n stands for the coefficient of fourier series. Then f(p)=0 whenever f is continuous at the point p.

2. Relevant equations

3. The attempt at a solution
assuming f is continuous at p=0, and supposing f(p)>0, the book (princeton lectures in analysis I) constructed trigonometric polynomials $$p_{k}(x)=(\epsilon+cos(x))^{k}$$ so that $$\int_{-\pi}^{\pi} f(x) p_{k}(x) dx$$ approaches infinity as k approaches infinity, contradicting the fact that the integral should* be zero. I have no idea why it should be zero. I thought c_n is defined to be $$\int_{-\pi}^{\pi} f(x) e^{-inx} dx$$ ? I do not see they are equivalent in some obvious way...what's the idea in it?
Thanks a lot

Last edited: Nov 17, 2009