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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?

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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.

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