# Fourier series convergence 1

For a square integrable function $$f$$ on $$[-\pi ,\pi]$$ fourier series of $$f$$ converges to $$f$$ in the sense $$\lim\limits_{n \rightarrow \infty} \| f - S_n \| = 0$$ where $$S_n(t) = \frac {a_0} {2} + \sum\limits_{k=1}^{n} (a_k \cos kt + b_k \sin kt).$$

Does this definition of convergence imply that $$f(x) = \text {fourier series of}\ f$$?

Does this definition hold for function which are not square integrable?

If $$f$$ is square integrable then the Fourier sereies of $$F$$ converges to $$f$$ in the sense the partial sums converge to $$f$$ in $$L^{2}$$ norm. This does not mean that the series converges for any particular value of $$t$$. For integrable functions which are not square integrable convergence of the series to $$f$$ is more complicated and it requires additioal assumptions.