with the help of Fourier series theory on $[-\pi,\pi]$, show that for any given periodic continuous function (complex valued) $f$ on $[-\pi,\pi]$ of period $2\pi$ and $\epsilon>0$, there exists $P=\sum_{|k|\le N} c_ke^{ikx},n\in\mathbb N,c_k\in\mathbb C,$ (for all $k$) such that $|f(x)-P(x)|<\epsilon$ for all $x\in [-\pi,\pi]$.
I know that for a Fourier series to converge uniform requires the function to be piecewise smooth or piecewise continuous and differentiable on $[-\pi,\pi]$. This question does not state that condition. Any idea how to solve the above question? thanks
You should phrase your question properly.It's not true that given any continuous map $f$ on the unit circle,it's Fourier series must converge uniformly.Infact,the Fourier series might not even converge pointwise.What you have asked for here,is a proof of the fact that every continuous $2\pi$-periodic map $f$ on $[-\pi,\pi]$ can be uniformly approximated by a sequence of trigonometric polynomials,which is a form of Weierstrass approximation theorem.The thing is,those $c_{k}'s$ might not be the Fourier coefficients at all.The standard way to prove this is by using Fejer summability of Fourier series.See below:-
https://en.m.wikipedia.org/wiki/Fejér%27s_theorem
Here,as you can see,the sequence of trigonometric polynomials converging uniformly to $f$ is given by the Cesaro means.Hope this helps.